Recent Developments in Stochastic Analysis and Related Topics: Proceedings of the First Sino-German Conference on Stochastic Analysis (a Satellite Conference of ICM 2002) 9789812561046, 981-256-104-8

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Recent Developments In Stochastic Analysis a n d Related Topics

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Proceedings of the First Sino-German Conference on Stochastic Analysis (A Satellite Conference of ICM 2002)

Recent Developments In Stochastic Analysis and Related Topics Beijing, China

29 August - 3 September 2002

editors

Sergio Albeverio Universitaet Bonn, Germany

Zhi-Ming M a Chinese Academy of Sciences, China

Michael Roeckner Universitaet Bielefeld, Germany

N E W JERSEY

-

K World Scientific LONDON

*

SINGAPORE * B E l J l N G * S H A N G H A I * HONG KONG

-

TAIPEI

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

RECENT DEVELOPMENTS IN STOCHASTIC ANALYSIS AND RELATED TOPICS Proceedings of the First Sino-GermanConference on Stochastic Analysis (A Satellite Conference of ICM 2002) Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-104-8

Printed by FuIsland Offset Printing (S) Pte Ltd, Singapore

Organizers Sergio Albeverio

Zhi-Ming Ma

Michael Rockner

Academic Committee Members E. Bolthausen A. B. Cruzeiro H. Follmer F. Gotze N. Krylov D. Nualart J.A. Yan

L. Accardi M. F. Chen K. D. Elworthy M. Fukushima L. Gross P. Malliavin G. Da Prato

Session Organizers P. Blanchard

A. Bovier Y. Kondratiev M. P. Qian M. Schweizer K. T. Sturm

N. Jacob R. Lkandre M. Schurmann B. L. Streit L. Tubaro

Sponsored By The Sino-German Center The National Natural Science Foundation of China

V

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Preface The present book consists of contributions given by the participants to the conference “The First SineGerman Conference on Stochastic Analysis (A Satellite Conference of the International Congress of Mathematicians 2002)”, held in Beijing 29. Aug. 2002 - 03. Sep. 2002, at the SineGerman Science Centre. The idea of the Conference was to present the state of the art and propose new developments in the area of stochastic analysis and its applications. It continues the tradition set by a conference initiated by the collaboration of S. Albeverio, Z.M. Ma and M. Rockner, having taken place in Beijing in 1993. (The Proceedings have appeared as Z.M. Ma, M. Rockner, J.A. Yan, ed Dirichlet Forms and Stochastic Processes, de Gruyter, Berlin, 1995) The present Conference was made possible by the generous support of the SineGerman Centre in Beijing. According to the rules of the Centre, a t least 80% of the participants had to have their working place in Germany or China. The organizers, S. Albeverio, Z.M. Ma, M. Rockner, and an international Scientific Committe consisting of L. Accardi, E. Bolthausen, M.F. Chen, A.B. Cruzeiro, K.D. Elworthy, H. Follmer, M. Fukushima, F. Goze, L. Gross N. Krylov, P. Malliavin, D. Nualart, G. Da Prato, J.A. Yan, selected the scientific topics of the Conference and the key speakers in each area, taking care of having, in addition to international experts, a number of excellent young lecturers. The realization of the project was sustained by a local Organizing Commitee, consisting of D.Y. Chen, M.F. Chen, F.Z. Gong, Z.M. Ma, F.Y. Wang, and J.A. Yan. The Conference was structured into 10 main areas, in which a total of 35 invited lectures were held. Corresponding to each area, sections were organized, with contributed lectures. The sections and their corresponding organizers were: (1) Geometry on path space: R. Leandre (Nancy) (2) Infinit dimensional analysis, measure-valued processes, and Dirichlet forms: K.T. Sturm (Bonn) (3) Quantum probability and related topics: M. Schiirmann (Greifswald) (4) Pseudo-differential operators and jump processes: N. Javii

viii

cob(Swansea) ( 5 ) Random media: A. Bovier (Berlin) (6) Statistical mechanics and particle systems: Y. Kondratiev (Bielefeld) (7) Stochastic finance: M. Schweitzer (Munchen), J. A. Yan (Beijing) (8) Stochastic metho in quantum field theory and hydrodynamics: Ph. Blanchard, L. Streit (9) Stochastic partial differential equations: L. Tubaro (Trento) (10) Markov processes and applications: M.P. Qian (Beijing) The present Proceedings reflect contributions of all these areas. Stochastic analysis and geometry on Riemannian manifolds and path spaces is discussed in contributions by S. Aida, A.B. Cruzeiro and X.C. Zhang, X.D. Li, K.D. Elworthy and F. Y. Wang, I. Mitoma, N. Sidorova, O.G. Smolyanov, 0. Wittich and H. V. Weizsacker. particular essential spectra, logarithmic Sobolev inequalities on pinned path spaces, Ornstein-Uhlenbeck semigroups on path spaces, Littlewood-Paley inequalities on manifolds and Brownian motion on submanifolds are investigated. Infinite dimensional analysis, measure-valued processes, Dirichlet forms are discussed in contributions by Z.Q. Chen and R. Song, U. Freiberg and J.U. Lobus, as well as in S. Liang, S.M. Li, Y. Ogura, F.N. Proske and M.L. Puri. Topics discussed include Green functions estimates, nodal sets, precise large deviations estimates, central limit theorems for generalized set-valued random variables. Quantum probability and quantum computing is discussed in contributions by S. Albeverio, S.M. Fei and X.H. Wang, U. Franz, Z. Huang and G. Rang, Y.G. Lu and S. Ruggeri, and M. Skeide. This includes topics like measures of entanglement, Levy processes Lie-algebras, quantum field theory, independence and product systems. Pseudo differential operators and jump processes are discussed by T. Komatsu and A. Takeuchi and R. Leandre, including topics Ii generalized Hormander theorems and Nash inequalities. Random media are discussed in a contribution by B.G. Ben Arous, L.V. Bogachev and S.A. Molchanov. In particular new limit laws of random exponentials are derived. Statistical mechanics and particle systems are discussed in contributions by H. Gottschalk, and H. Ouerdiane and L. Silva. This includes the discussion of new ferromagnetic systems and transport equations.

ix

Economics and finance are discussed in contributions by M.F. Chen on stochastic models of economic optimization, and by A.S. Popovici, on forward growth rates in finance. Stochastic methods in quantum field theory and hydrodynamics are discussed in contributions by S. Albeverio and B. Ferrario, Z. Brzezniak and Y.H. Li, as well as by J. Lorinczi. This includes the discussions of invariant Gibbs measures for fluida, stochastic Navier-Stokes equations as well as of ground states in Nelson’s model of quantum fields in interaction with non relativistic particles. Stochastic partial differential equations are discussed by D. Blomker and C. Gum, and by S.V. Lototsky. This includes results on thin-film-growth models and optimal filtering for stochastic parabolic equations. The Conference and its proceedings put in evidence the strong interconnection of stochastic analysis with other areas of mathematics, as well as with applications of mathematics in natural and socio-economic sciences. The strong ties between Chinese and German groups working in these areas have been greatly enhanced by the Conference. We are very grateful to the National Science Foundation of China(NSFC), the 973 Project of the Chinese Ministry of Sciences and Technology, the German Science Foundation (DFG), the Bielefeld-BonnStochastics Research Centre (BiBoS), the SFB 611 (Bonn) and the DFG Forschergruppe “Spectral anlysis, asymptotic distributions and stochastic dynamics” for financial support. The success of the Conference was made possible through the precious help of Prof. Dr. D.Y. Chen (Beijing University), Dr. B. Nunner (DFG, Bonn), Prof. Dr. F.Y. Wang (Beijing Normal University) and Dr. M.G. Zhao (SineGerman Science Centre, Beijing). Our special warm thanks go to Mei-Ling Wang, who through her competence both in German and Chinese Science organizations, and her strong personal engagement managed to solve the numerous problems which arose during the organization of the Conference. Without her help the Conference would never have been realized. We are also grateful to the coworkers of the members of the local committee in Beijing and of the Mathematics Institutes in Bielefeld (Prof. M. Rockner) and Bonn (Prof. S. Albeverio), wh helped in solving practical problems in connection with the Conference. For the preparation of the Proceedings we are very grateful to Judith Dohmann, who took all burden to collect all contributions and carry out the

X

correspondence with referees (all papers have been internationally refereed). Thanks also go to the coworkers of Ping-Ji Deng for the 7&X-related work in connection with the setting of the book for publication. Beijing, Bielefeld, Bonn Sergio Albeverio

October 1, 2003

Zhi-Ming Ma

Michael Rockner

List of Participants Accardi, Luigi Assing, Sigurd Bismut, Jean-Michel Bogachev, Leonid Bovier , Anton Chen, Chuanzhong Chen, Jinwen Chiang, Tzuu-Shuh Cipriani, Fernanda Deng, Pingji Dong, Zhm Elworthy, David Fang, Shizan Farkas, Erich Walter Fleischmann, Klaus Freudenberg, Wolfgang Gong, Guanglu Grothaus, Martin Hahn, Atle Hesse, Martin Hou, Zhentin Hu, Zechun Ji, Hui Jiang, Yiwen Kondratiev, Yu. Kuwae, Kazuhiro Lkandre, R6mi Li, Chujin Li, Xiang Dong Liang, Song Liu, Cihua Liu, Juxin Liu, Yong Lu, Yungang Lytvynov, E. Mao, Yonghua Mitoma, Itaru

Aida, Shigeki Banulescu, Martha Blanchard, P. Bogachev, Vladimir Brzezniak, Z. Chen, Dayue Chen, Mufa Chow, Yunshyong Cruzeiro, Ana Bela Deuschel, Jean-Dominique Du, Yonghong Fan, Longzhen Fang, Xing Feng, Shui Fkanz, Uwe Gao, Fuqing Gotze, Friedrich Guan, Qingyang He, Kai Hoh, Walter Hsu, Elton Pei Huang, Zhiyuan Jiang, Daquan Klein, M. Kiilske, Christof Lan, Guolie Lei, Liangzhen Li, Ping Li, Yuhong Liang, Zhongxia Liu, Guoxin Liu, Lixin Liu, Zaiming Luo, Jiaowan Ma, Yutao Merkl, Franz Ndumu, Martin N. xi

Alb everio, Ser gio Bao, Ying Blomker, Dirk Bolthausen, Erwin Cao, Guilan Chen, Jiaqing Chen, Shuang Cipriani, Fabio Deng, Aijiao Dohmann, Judith Eberle, Andreas Fan, Xiaoming Fei, Shao-Ming Ferrario, Benedetta Freiberg, Uta Gong, Fuzhou Gottschalk, Hanno Guo, Junyi He, Ping Hopfner, R. Hu, Yijun Imkeller, P. Jiang, Wenjiang Komatsu, T. Kuna, Tobias Lao, Lanjun Li, Bo Li, Shoumei Li, Zenghu Linde, Werner Liu, Jicheng Liu, Yan Lorinczi, J ozsef Luo, Shunlong Ma, Zhi Ming Miao, Yu Nunner, Bernhard

xii

Oliveira, Maria Joao Popovici, Stefan Alex Rang, Guanglin Von Renesse, Max-K. RUSSO,F'rancesco Schurmann, Michael Sinha, K.B. Song, Qiongxia Sturm, T. Teichmann, Josef Ugolini, Stefania Wang, Fengyu Wang, Lie Wei, Li Wu, Liming Xia, Jiaming Xie, Peng Yan, Jia-An Yang, Zhihong Zambotti, Lorenzo Zhang, Fuxi Zhang, Lihong Zhang, Tusheng Zhang, Xinsheng Zhao, Huaizhong Zhong, Yuchuan

Ouerdiane, Habib Posilicano, Andrea Redina, Victoria Rockner, Michael Schmuland, Byron Shi, Jimin Skeide, Michael Song, Shiqi Streit, Ludwig Thalmaier, A. Wang, Caishi Wang, Guilan Wang, Meiling Weizsacker, Heinrich V. Wu, Rong Xie, Bin Xie, Yingchao Yang, Haicheng Ye, Zhongxing Zahle, M. Zhang, Huizeng Zhang, Mei Zhang, Weiyi Zhang, Yongfeng Zhao, Minzhi Zou, Jiezhong

Ouyang, Shunxiang Qian, Minping Ren, Jiagang Rudiger, Barbara Schott, R. Signahl, Mikael Sobol, Z. Stannat, Wilhelm Su, Zhonggen Tomisaki, Matsuyo Wang, Feng Wang, Jixia Wang, Yingzhe Wu, Jiang-Lun Wu, Ying Xie, Jiansheng Yan, Guojun Yang, Weiguo Ying, Jiangang Zhang, Bao Zhang, Jingxiao Zhang, Na Zhang, Xiaomin Zhang, Yuhui Zhao, Xuelei

Content Preface

vii

List of participants

xi

Precise Gaussian estimates of heat kernels on asymptotically flat Riemannian manifolds with poles Shig e ki A ida

1

Equivalence of bipartite quantum mixed states under local unitary transformations 20 Sergio Albeverio, Shao-Ming Fei and Xiao-Hong Wang Invariant Gibbs measures for the 2D vortex motion of fluids Sergio Albeverio and Benedetta Ferrario

31

Limit laws for sums of random exponentials Ge'rard Ben Arous, Leonid Bogachev and Stanislav Molchanov

45

Thin-Film-Growth-Models: On local solutions Dirk Blomker and Gugg Christoph

66

Asymptotic behaviour of solutions to the 2D stochastic NavierStokes equations in unbounded domains - new developments 78 Brzeiniak Zdzistaw and Yu-Hong La Stochastic models of economic optimization Mu-Fa Chen

112

A note on the Green function estimates for symmetric stable processes 125 Zhen-Qing Chen and Ren-Ming Song Ornstein-Uhlenbeck semigroups on Riemannian path spaces 136 Ana Bela Cwzeiro and Xi-Cheng Zhang Essential spectrum on Riemannian manifolds K. D. Elworthy and Feng-Yu Wang xiii

151

xiv

LQvy process on real Lie algebras

166

Uwe Franz Zeros of eigenfunctions of the generator of a gap diffusion

182

Uta Freiberg and Jorg-Uwe Lobus Wick rotation for holomorphic random fields

199

Hanno Gottschalk White noise approach to interacting quantum field theory

220

Zhi- Yuan Huang and Guang-Lin Rang Generalized Hormander theorem for non-local operators

234

T. Komatsu and A. Takeuchi Stochastic mollifier and Nash inequality

246

Rkmi Le'andre Precise estimations related to large deviations

257

Song Liang Central limit theorems for generalized set-valued random variables 271

S. La, Y. Ogura, F.N. Proske and M. L. Puri Littlewood-Paley-Stein inequalities and Riesz transforms on complete Riemannian manifolds 289

Xiang-Dong L i The ground state in Nelson's model with or without infrared cutoff 309

Jdzsef LBrinczi Optimal filtering of stochastic parabolic equations

330

s. v. Lototsky A new example of interacting free Fock space Y.G. L u and S. Ruggeri

354

Stochastic holonomy

370

Itaru Mitoma

xv

On the stochastic transport equation of convolution type Habib Ouerdiane and Jose' Luis Silva

384

Forward growth rates: connecting the HJM and the BS mod399 els Alex Stefan Popovici Independence and product systems Michael Skeide

420

Brownian motion close to submanifolds of Riemannian manifolds 439 N . Sidorova, 0. G. Smolyanov, H. V. Weizsacker, 0. Wattich

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Precise Gaussian estimates of heat kernels on asymptotically flat Riemannian manifolds with poles Shigeki Aida Department of Mathematical Science, Graduate School of Engineering Science Osaka University, Toyonaka, 560-8531, JAPAN e-mail: [email protected]

Abstract We give precise Gaussian upper and lower bound estimates on heat kernels on Riemannian manifolds with poles under assumptions that the Riemannian curvature tensor goes to 0 sufficiently fast at infinity. Under additional assumptions on the curvature, we give estimates on the logarithmic derivatives of the heat kernels. The proof relies on the Elworthy-Truman’s formula of heat kernels and Elworthy and Yor’s observation on the derivative process of certain stochastic flows. As an application of them, we prove logarithmic Sobolev inequalities on pinned path spaces over such Riemannian manifolds.

1. Introduction Let p ( t , z, y) be the heat kernel of a diffusion semigroup etA12 on a ddimensional complete Riemannian manifold ( M ,g), where A denotes the Laplace-Beltrami operator on ( M ,9). For some classes of Riemannian metrics, the following Gaussian upper and lower bound are valid (see [31]): For all t > 0, x,y E M , it holds that

where d ( z , y) denotes the Riemannian distance between x and y, C1, C3 are positive constants and C2, C, are nonnegative constants with Cz < 1. It is natural to expect that more precise estimate holds under stronger 1

2

assumptions on the Riemannian metric. In fact, under nonnegativity of the Ricci curvature, Li-Yau’s lower bound estimate [25], (1.1) with Cl = ( 2 7 ~ ) - ~Cz / ~= , 0, holds. Also if the sectional curvature is nonpositive, the upper bound in (1.1)holds with Cs = (27~)-~/’ and C, = 0 by [8]. Note that the lower bound in (1.1) does not hold for t E [O,T]for any fked T in the case of hyperbolic spaces. In [2], the author proved the lower bound with C2 = 0 for any t E [O,T]under the assumptions that x is a pole and the derivatives of Riemannian metric go to 0 sufficiently fast at infinity although negative curvature part remains. In this paper, we prove similar estimates on Riemannian manifolds with poles under the assumptions that the curvature and the derivatives go to 0 sufficiently fast at infinity. The present proof is simpler than the previous one. Also we discuss estimates on the logarithmic derivatives of heat kernels. The second derivative of logp(t,x,y) with respect to y was studied in connection with parabolic Harnack inequality [25], [33]. On the other hand, it is pointed out in [19], [23] that V i logp(t, x, y) is related to a logarithmic Sobolev inequality (=LSI for short) on loop space over the Riemannian manifold. In [l],sufficient conditions for LSI in terms of the heat kernel were given. Malliavin and Stroock [26] showed that the small time behavior of the second derivative of logp(t, z, y) dramatically changes if y is in the cut-locus of 2. However in our case, there are no cutlocus and we can check the conditions in [l]by our main thorem. The key ingredients of our arguments are the Elworthy and Truman’s formula [16] and Elworthy and Yor’s observation on the derivative process of stochastic flows [17]. In [30], Ndumu studied derivative formulae of heat kernels by using Elworthy and Truman’s formula. However, his assumptions on the boundedness of the derivative process is too restrictive. The key point in the present paper is to use Elworthy and Yor’s observation to avoid the difficulty. Acknowledgement: I am grateful to David Elworthy for informing me a preprint of Ndumu [30]. Also I thank the referee for the comment on the heat kernel bound.

2. Elworthy and Truman’s formula and estimates on heat kernels

Let ( M ,9 ) be a d-dimensional complete Riemannian manifold with a pole 0. That is, we assume that the exponential map exp : TOM ---t M is diffeomorphism. Also we assume that ( M ,9 ) is a stochastically complete,

3

that is, the heat kernel satisfies J M p ( t , z , y ) d y = 1 for all z , y E M . Let (2.1) Note that 8 is a positive smooth function on M . This is called a Ruse’s invariant. Now, we embed M into a higher dimensional Euclidean space RN isometrically and let P ( z ) : RN + T,M be the projection operator. Let us consider the following SDE with a singular drift at time t :

d X ( s ,z, W ) = b, ( X ( S 2, , w ) )ds

+ P ( X ( S 2,, w ) ) o d w ( s) (0 5 s < t ) (2.2)

X ( 0 ,z, w ) = 2,

(2.3)

where

For simplicity, we denote E ( z ) = d(o,z)’/2 for sometimes. ElworthyTruman [16], [13] proved that

Theorem 2.1.

(1)

where

1

V(z) = -8(z)1/2A (8-’l”) 2

(x).

(2.7)

(2) The law of the process d(o, X ( s ,2,w ) ) is the same as that of the radial (O) part r d ( o , z ) ( t ) of the pinned Brownian motion o n Rd such that T ~ ( ~ , ~ ) = d(o, x) and ? - d ( o , z ) ( t ) = 0, that is, the pinned Bessel process with dimension d . I n particular, lim,,t X ( s ,z, w ) = 0. We consider the following assumption. We identify a second covariant derivative of a function with a symmetric operator below.

Assumption 2.2. ( A l ) The n-th covariant derivatives of logB(x) (1 5 n 5 4) are bounded continuous functions o n M .

4

(A2) There exists a positive constant E > 0 such that for all x E M ,

02

d(o,x)2 {F }2 ~1 I T , M .

(A3) There exists a constant C

+E

> 0 such that for all x

E

M,

(A4) Ricci curvature and its first derivative are bounded. By using the Levi-Civita connection, the semimartingale X ( s ,x , w ) (0 5 s < t ) can be lifted to the orthonormal frame bundle O ( M ) and stochastic parallel translation T ( X ) , : T,M 4 T X ( , , ~ , ~can ) Mbe defined. Any tensor T on T x ( t , Z , w ) M can be parallel translated to a tensor T ( X ) ,on T x M along X . We use this notation frequently. The following derivative formulae are keys in our argument. The formulae (2.10) and (2.11) hold under weaker assumptions. However, we do not intend to refine the results in this paper.

Lemma 2.3. Assume ( A l ) , (A.2)) (A3)) (A4). Then the following formulae hold:

5

TxM* @ T x M are the solutions to the following ODES:

(2.13)

VZ(c.1,

c.270) = 0.

Moreover, V, VV, V2V and S U P O < ~ < ~ Il~~(t)llop, SUP^+^^ l l ~ 2 ( ~ ) l l o p are bounded functions of x and w. In paGicular, Vxh(t,0,x),Vzh(t, 0,x) are bounded on the whole space M . Proof. Note that

v ( ~=)-41 (

I V ~ ~ ~ B -2 ( ~Z i) ~~g~e ( ~ ) ) .

Hence, ( A l ) implies V, VV, V2V are bounded. Let N ( x ) : Rd -+ TxM' be the projection operator. We also fix a Riemannian connection on the normal bundle 7r : N ( M ) -+ M . Let

X ( s )=

I'

.(X),

0

(2.14)

dX(s, 2 , w)

(2.15) (2.16) Note that b(s) and p(s) are independent Brownian motions on T x M and T x M L respectively. Since X ( s ) satisfies the following SDE, 1dX(s) = --VE(X),ds t-s X(0) = 0,

1 2

- -V logB(X),ds

+ db(s),

(2.17) (2.18)

6

X(s) and X ( s , x , w ) are %-measurable, where % = a ( b ( s ) I 0 5 s < 1). Note that, formally, q ( E , s) = ~(x);'&X(s, z, w)(E) satisfies the following SDE:

Vl(E,O)

= E,

where A denotes the shape operator of M in RN. The conservativeness assumption on X implies that X ( t ,x,w) exists for all t > 0. However the differentiability problem on x is not easy. In the case of compact manifolds, there is a version such that for almost all w and all t > 0, IC -+ X ( t , z , w ) is a diffeomorphism and (2.19) holds. See [17], [4]. Roughly speaking, (2.12) can be obtained by taking the conditional expectation of q ( E , s ) with respect t o % by noting the independence of % and ,B. Similarly (2.10) can be proved by taking the conditional expectation with respect to '13 in the Wiener functional representation formula for V,h(t, o, z) which is obtained by taking the derivative (2.6) with the help of (2.19) and (2.12) as in [17], [4], [14]. However A(X), may not be integrable function on the Wiener space and so we should be careful to take derivative and the conditional expectation. Hence, we need consider the approximate function of h(t,o, x) to differentiate itself. Let h€,L(t,0,).

(2.20)

Here ' p ~ is a smooth cut-off function whose support is in [-L2, L2] and p(u) = 1 for u E [-L,L] and all derivatives of ' p goes ~ t o 0 uniformly on R when L --f co. 11 Iln,m,t is given by

< ~t< 1/2,21tm > 1 and m is an integer. Note that the norm can be defined for positive real number m satisfying the above relation on m and K . Then lima--tO,L+mh E , ~ o, ( tx) , = h(t,o, z). In (2.20), we may assume X ( t , x , w ) is smooth with respect t o x because we may

where 0

11

IIK,m,t

7

assume X(., z, w)moves in a compact subset thanks to the existence of the cut-off function. Thus we can differentiate both sides of (2.20) and we may assume that the equation (2.19) is valid up to the exit time of X(t,z, w) from a compact set. Consequently, we have

(Vk,N(t,0,x),r>

(2.21)

If X(s,z,w) moves in a compact subset in (2.21), (2.22), A(X), are bounded and so it holds that for all -15 p < 00 and 0 < 1 < 00, r

1

Therefore S U ~ 1 5 p < oo. Since

WE,

W)

~

Ilvl( 1. By using these estimates, the proof in Theorem 2.4 works. We omit the details. 0 Let P,,,(M) = C([O,l] M I $ 0 ) = z , y ( l ) = y) and we denote the pinned Brownian motion measure by v,,~. On P z , y ( M ) ,a Dirichlet form is naturally defined by the H-derivative D on P z , y ( M ) .See [l].Let ZCF

{*}>

--f

14

be the set of all smooth cylindrical functions on P z , y ( M ) . The following inequality is called a logarithmic Sobolev inequality( =LSI): There exists C > 0 such that for all F E ZCF

(2.62) When M is a Euclidean space, (2.62) holds witb C = 2 by Gross' result [18]. Driver and Lohrenz [9] proved LSI on loop group for the heat kernel measure which is equivalent to the pinned Brownian motion measure [lo, 31. On the other hand, Eberle [ll]proved that PoincarB's inequality does not hold on a loop space with pinned measure over certain simply connected compact Riemannian manifold. Therefore, LSI does not hold in such a case. But the validity of LSI for pinned measure is still an open problem generally. In the case of Riemannian manifolds with poles, we can prove the following.

Theorem 2.6. (1) Assume that (2.27), (2.29) with 6 < 1/3 and (2.35) hold. T h e n (2.62) holds in the case where x = o and f o r all y . The constant C depends only o n a and 6 . (2) Assume (2.37), (2.39), (2.41) and (2.44). T h e n (2.62) holds f o r any z and y . Proof. This follows from Theorem 3.6 in [l]immediately.

0

3. Rotationally symmetric case In this section, we consider rotationally symmetric Riemannian metric. We fix an orthonormal frame {ei}:=l c TOMand identify TOMwith EXd. Let @ : EX+ x Sd -+ TOMbe the natural map, @ ( r , w )= r w , where r = d ( o , s ) , x = exp(rw), ( r 2 0, w E S d - l ) , Sd-' is the unit sphere centered at the origin in T O M .g is called a rotationally symmetric if the pull back of g by @ can be expressed as

(@ exp)*g = dr2 + f ( r ) 2 d w , +

dw denotes the standard Riemannian metric on the sphere. We introduce p ( r ) by f ( r ) = r e p ( T ) .Then by the definition of 8,

Lemma 3.1.

15

Under the assumption of the rotationally symmetry, there exists a smooth function of r , p ( t , r ) such that p ( t , 0,z) = p ( t , d(o, z)). Note that f is a C" function on [0, cm) satisfying f (0) = 0, f ' ( O ) = 1 ( [2O]). Since f'(0) = 1, note that 'p(0) = 0. Let K ( r ) be the radial curvature at x. Then the following Jacobi equation holds (see page 30 in [20]). f"(r) = -K(r)f (r).

(3.3)

In Section 2, we have given estimates on heat kernels under assumptions on the Riemannian curvature. In this section, we will give similar type estimates on heat kernels in terms of 'p(r). In rotationally symmetric case, we can go further than general cases. To explain it, let us consider the hyperbolic space with constant negative curvature. In that case, it holds that for any fked T > 0,

where f ( t ,x) is defined in (2.33) although inf, f ( t ,x) = 0 which is excluded under the assumption (2.31). In rotationally symmetric case, we can prove (3.4) under an assumption (Assumption 3.2) which is valid for hyperbolic space. Of course, the similar estimate should hold without rotationally symmetry under suitable assumptions. We study this in future papers. We use the following assumption on 'p.

Assumption 3.2. The k-times derivative 'p(')(r) is a bounded function on [0,co) for all k 2 1. Moreover there exists a C" function 4 o n [0,cm) such that p ( r ) = 4(r2).

Remark 3.3. (1) W h e n M is the hyperbolic space with sectional curvature --a, then K ( t ) G --a and f ( r ) = 'T-. sinh f i r Thus (3.5)

where we write subscript a to denote the dependence of the curvature. Since the following Taylor expansion holds f o r all r 2 0, sinh J ..

J.. 'pa(&)

" (ar)"

z l + C( 2 n + 1 ) ! ' n=l

is a smooth function on [ O , c m ) . Also 1 'pi(r)= cothr - -. r

(3.6)

16

It is easy to see that this function and its all derivatives are bounded functions on [0,cm). Therefore Assumption 3.2 holds for hyperbolic spaces. (2) By the Jacobi equation, we have (3.7) =

-

+

+

(4r24’(r2)2 64’(r2) 4r24”(r2)) .

Therefore, b y the lemma below, under Assumption 3.2, it holds that sup IK(r)l < 00. T>o

Lemma 3.4. Under Assumption 3.2, for any k 2 1,

Proof. We will prove this by induction on k. Because (p(r)= q5(r2), (p’(r)= 2 r 4 ’ ( r 2 )holds. Since is a bounded function, we have d / ’ @ ( r ) is also bounded. We assume that up to k the assumption of induction holds. Taking k 1-times derivative, we have

+

= p r ) k + 14 @+‘I(r 2 ) + G k ( r ) .

Here G k ( r ) is the sum of the function rm$(‘)(r2),where nonnegative integers m and 1 satisfy that m < 1 < k 1. Hence by the assumption of induction, G k ( r ) is a bounded function on [0,cm). Noting that (p(”’) is 0 also bounded, we see that induction is completed.

+

Lemma 3.5. (1) Let F be a C2-function on R. Then we have

(t + ) P’,

V a F ( r )= F‘(r) -

(p’(r)

+ F”(r)PX. ( r # 0 ) ,

(3.9)

where P, denotes the projection operator onto the 1-dimensional subspace in T,M spanned b y v E T x M where exp,v, = o and Pk denotes the orthogonal projection. Note that we identify the symmetric bilinear form and the symmetric map on T,M through the metric. (2) The first and second covariant derivatives of log e(z) = ( d - l)cp(r) are bounded functions on M . (3) When f ( r ) = re+’(T), it holds that d-1 V(Z) = -( ~ ” ( r ) ( d - 1)r 4

+

+d 2 1 -

(3.10)

Assume Assumption 3.2. Then V, VV, V2V are hounded functions on M .

17

Proof. (1) follows from a formula in page 30 in [20]. (2) follows from 0 the definition and (1) and Lemma 3.4. Theorem 3.6. Assume Assumption 3.2. We have the following explicit expression and a n estimate for the logarithmic Hessian of the heat kernel.

-

(: +

‘p’(r))e p 2’(r)P:

+ V:

log h(t,0 , z),

and for any T > 0 , sup

/IV,logp(t,o,x) -

< co,

(3.11)

Z€M,O

1 be

(2.2)

E R1/, and put

Then 1

h(s)

N

e V+(X)

(X + W)

(2.4)

if and only i f H(t)

N

1 7$'(t) =: Ho(t)

e

(t -+ w).

(2.5)

In particular, if h E Re then H E Re!, and vice versa. In the sequel, the following simple identity will be useful, which is just a rearrangement of the definition (2.2):

e' e

- = p'-

1.

Note that the expected value of the sum S N ( ~is) given by N

E[,S"(t)] =

E[etxi] = N e H ( t ) , i=l

suggesting that the function H ( t ) sets up an appropriate (exponential) scale for N = N ( t ) . In fact, it is technically more convenient of the form t o use Ho(t) as a rate function [see (2.5)].2 More precisely, denote log N X := liminf t-cc Ho(t) and set A1

:= e' - 1,

A2

:= 2Q'(p' - 1).

(2.8)

This makes no difference in the 'crude' Theorems 3.1 and 3.2 below, since H o ( t ) N H ( t ) as t -t 00, but it will be crucial for the more delicate Theorems 5.1,5.3 and 5.4.

52

These two values prove to be critical ones with respect to the scale (2.7). Let us also introduce a new parameter,

a = a(e,X) :=

($)

lle'

Conversely, in view of formula (2.6) the parameter X is expressed through aas

x = aQ'(p'-l),

(2.10)

and from (2.8) it follows that the respective critical values of a are given by a1

=1,

a2

=2.

We will see that a plays the role of characteristic exponent in the limit laws.

3. 'Crude' limit theorems above the critical points Our first theorem asserts that if N grows fast enough then S N ( ~satisfies ) the Law of Large Numbers in its conventional form.

Theorem 3.1 (LLN, X

> XI).

Suppose that X

> XI, and set

Then

s;(t) -51

(t + m).

Proof. It suffices to prove that for some r > 1 lim E[S;\r(t) - 11'

t-icc

= 0.

By an inequality of von Bahr and Esseen [(1965), Theorem 2, page 3011, for any r E [l,21

EIS;

- 11'

+

5 2N1-' EletX-H(t) 11'.

+

+

Furthermore, by the elementary inequality (z 1)' 5 2'-'(zT 1) (z 2 0, r > I), which easily follows from Jensen's inequality applied to z', we get

E[S;, - 11' 5

2"N1-'eH(Tt)-TH(t)

+ 0(N1-').

(3.1)

53

Since H E Re, and also using (2.7) and the asymptotic equivalence H ( t ) Ho(t) [see (2.5)], we obtain

= X(r - 1) - re'

-

+ r =: .A(.).

Note that v ~ ( 1 = ) 0 and vi(1) > 0 [due to the condition X > A1 = p' - 1, Hence, there exists r > 1 such that .A(.) > 0, which implies see (2.8)]. 0 that the exponential term in (3.1) is bounded by e-cH(t) = o(1). Our next result concerns the fluctuations of the sum S,(t) about the expected value. Note that Var[etx]= e H ( 2 t ) - e2H(t) Theorem 3.2 (CLT, X > A,).

-

(t 4 CQ).

eH(2t)

Suppose that X

> X2. Then

Proof. By the Lyapunov theorem [ see Petrov (1995), Theorem 4.9, page 126 1, we only need to check that for an appropriate r > 1, N

N-' e--rH(2t)

EletXi - eH(t)12r4 0

(t --t a).

(3.3)

i= 1

Arguing as in the proof of Theorem 3.1, one can show that the left-hand side of (3.3) is dominated by 22r-1N1--re--rH(2t)

(

eH(2't)

+

e2rH(t))

-

22T-1N1-TeH(2Tt)-TH(2t)

Furthermore, analogously to (3.2) we obtain lim inf

(.(

N

g);;;-

H(2rt) H(t)

-~

t-im

= X(r

-

+=)H ( t )

1) - (a?-)@'+r2Q'

= 2"(2-e'X(r

-

1) - r e ' + r )

= 2Q'vy(r),

where A' := 2-Q'X > 2-Q'Xz = p' - 1 [see (2.8)] and the function .A(.) is defined on the right-hand side of (3.2). Similarly as above, there exists a 0 number r > 1 such that v ~ ~ ( r>)0, and hence (3.3) follows.

54

4. Normalized regular variation

Below the critical points, the behavior of the sum S,(t) becomes increasingly sensitive t o subtle details of the upper tail's structure. So to get enough control on these, we require slightly more regularity of the distribution tail.

Regularity Assumption 2. The log-tail distribution function h is normalized regularly varying (with index 1 < e < m). The latter means that for each E > 0 the function x - Q f a h ( x )is ultimately increasing, whereas the function x-Q-'h(z) is ultimately decreasing [cf. Bingham, Goldie and Teugels (1989), Section 1.3.2, page 151. More insight into the property of normalized regular variation is given by the following lemma [cf. Bingham, Goldie and Teugels (1989), Section 1.3.2, page 151.

Lemma 4.1. A positive (measurable) function h i s normalized regularly varying with index p if and only i f it i s differentiable (a.e.) and

xh'(x) lim -- e. h(x)

I-oc)

If the log-tail distribution function h is known to be normalized regularly varying (with index e), it follows that the function Ho, an asymptotic version of H provided by Kasahara's theorem [see (2.5)], is normalized regularly varying as well (with index e'). Moreover, by comparing equations (2.4) and (2.5) via the relation (2.3) and using that the ordinary inverses of the functions h ( x ) and Ho(t) exist (for large enough x and t ) ,one can show that for all t large enough, Ho(t) is the unique solution of the equation

Let us consider two examples to illustrate the difference between the functions H and Ho.

Example 4.2 (Weibull's distribution). Let X have the Weibull distribution,

P(X > x) = exp(-ze/e),

3:

2 0,

(4.3)

with p > 1. Then the log-tail distribution function reads h ( x ) = x Q / e and the density function is of the form fx(x) = x Q - l exp(-zQ/e) (x >_ 0). We

55

have eH(t)= E[etx] =

I”

xeP1exp(tx - xe/p) dz (4.4)

00

= te’

ye-l exp{ te’(y - ye/@)}dy,

where we used the substitution x = tQ’-ly and relation (2.6). Note that the function g(y) = y -ye/@ has a unique regular maximum at point y = 1, with g(1) = 1 - l/e = l/e’, g’(1) = 0, g”(1) = 1 - e < 0. Then the asymptotic Laplace method yields

whence

te’

e’

H(t)= e‘ + 2 logt

1 + -log 2 (@Z -1 + )o(1)

(t +.).

(4.5)

On the other hand, equation (4.2) can be easily solved t o obtain [cf. (4.5)1

te‘

t 2 0.

Ho(t) = -,

e’

Example 4.3 (Normal distribution). Let X have the standard normal distribution N ( 0 , l ) . Here e = e’ = 2, XI = 1, A2 = 4, and a: = A. The function h is given by h(x) = -log

(-&

e-y2/2dy)

2 2 +logz+- 1% -2 2

+

(. 1)

(x -m) i

and can be shown t o be normalized regularly varying. Note that for each tER E[etx] = et2l2, whence t2 t E R. 2 Equation (4.2) for HO can be solved asymptotically. For X $ {Al,X2}, one only needs to find Ho(t) to within o ( l ) ,

H ( t ) = log E[etx] = - ,

1 2

Ho(t) = - - log t - - log(27r) + o(1) t2

2

(t + m).

56

The case of the critical points is more subtle but is perfectly tractable as well. Let us now derive the most important implication of normalized regular variation, that is an exact identity relating the functions h and Ho. For 5 > 0, set

where p = p ( t ) is a (unique) solution of the equation

In particular, for z

=

1 we have

Using that h E R, and comparing the asymptotics of both parts of equation (4.8) as t 4 00, with the help of relations (2.2) and (2.9) we arrive a t the following assertion.

Lemma 4.4. The function p ( t ) has the limit

ex

lim p ( t ) = - . a!

t-00

(4.10)

Note that equation (4.8) combined with (4.2) yields

Hence, recalling the relation (4.9) we obtain our main result in this section.

Lemma 4.5 (Basic Identity). identity is true:

For all t large enough, the following

How the function v z ( t )emerges and the role of the Basic Identity will be explained later on.

57

5. Limit theorems below the critical points

In addition to regularity, more accuracy is now needed in specifying the rate of growth of N . Henceforth, we impose the following Scaling Assumption. The number N = N ( t ) of terms in the sum S,(t) satisfies the condition tJim -cc

Nexp{-XHo(t)}

where X is a parameter such that 0

= 1,

(5.1)

< X < 00.

Theorem 5.1 (Convergence to a stable law, 0 that 0 < X < X2, i.e. 0 < a < 2. Set

< X < A2).

Suppose

B ( t ) := exP{P(t)Ho(t)}

(5.2)

and

(5.3) I

0

if

0 < x < X I ( 0 < a < l),

where B1 ( t ) is a truncated exponential moment,

B l ( t ) := E[etxqx5,,(t)]].

(5.4)

Then

where Fa is a stable law with characteristic exponent a = a ( @A), defined in (2.9) and with skewness parameter ,8 = 1. The characteristic function & of the law 3ais given by

-r(i 1%

4 a).(

=

-

a)1uIaexp

(2 7 --

sgnu)

+

1 i sgn u . - log 1 u I 7T 2 ,

( f f

#

1)

( a = I)

where r(s) = J ~ x C s -e --“d l x is the gamma-function, sgnu := u/IuI for u # 0 and sgnO := 0 , and y = 0.5772.. . is the Euler c ~ n s t a n t . ~ See Gradshteyn and Ryzhik (1994), 8.367, page 955.

58

Remark 5.2. The scaling relation N

-

exp{XHo(t)} implies

B(t) = exP{P(t)Ho(t)}

NP ( t ) / X ,

By Lemma 4.4, we have

Hence, N is being raised to the power

This should be compared t o the classical results in the i.i.d. case [see, e.g., Ibragimov and Linnik (1971), Theorem 2.1.1, pages 37, 461, where the normalization is essentially of the form N1/*. As we see, the sums of random exponentials (1.1) have the limit distribution by virtue of a nonclassical (heavier) normalization. Let us now describe what happens at the critical points. In fact, the Law of Large Numbers and the Central Limit Theorem prove t o be valid at the critical points XI and X 2 , respectively; however the constants now require some truncation.

Theorem 5.3 (LLN, X = XI).

If X = X1 ( a = 1) then

where Bl(t) is given by (5.4).

Theorem 5.4 (CLT, X = X2). If X = A2 (a = 2) then

where Bz(t) is a truncated exponential moment of 'second order',

B2(t) := E[e2txl{x5?71(t)}].

(5.5)

6. Model example: Weibull's distribution For illustration purposes, let us give more explicit versions of the above limit theorems in the particular situation where X has the Weibull distribution (4.3).

59

Example 6.1 (Weibull’s distribution revisited). As shown in Example 4.2, in the Weibull case we have Ho(t) = te’/@’ [see (4.6)]. It is then easy to verify that the function p ( t ) , the root of equation (4.8), is given by

[cf. (4.10)]. So, from (5.2) using (2.9) we get

Furthermore, according to (4.9) we have q1(t) = (at)Q‘-l.

If Q = = 1 then (6.2) yields ql ( t ) = te’-l, so for the function B,(t) defined in (5.4) we obtain similarly t o (4.4) teI-1

zeP1exp(tz - xe/e) dx

Bl(t) =

(6.3)

1

= terJo

ye-’ exp{te‘(y - ye/@)) dy.

As shown in Example 4.2, the function g(y) = y

- ye/@ has a regular maximum a t point y = 1, which happens to be the right endpoint of the integration interval in (6.3). Hence, the Laplace method implies that, asymptotically, Bl(t) makes up exactly one-half of the full integral [cf. (4.4)]

that is to say,

Bl(t)

-

1 ,E[etx]

(t 4 m).

Similarly, from (6.2) with a = a2 = 2 we have q l ( t ) = (2t)e’-l. Hence, the function B2(t) defined in (5.5) is represented as (2t)Ql-I

xe-1 exp(2tx - x e / p ) dz

~ ~ = ( t )

(6.4)

1

=

( 2 t ) e . I y~-1exp{(2t)~’(y - ye/@))dy 0

[via the substitution z = (2t)Q’-’y], and exactly the same argument as before shows that 1 1 B2(t) 5 E p ] 5 V a r [ P ] .

-

-

60

As a result, we can combine the LLN of Theorems 3.1 and 5.3 as follows:

If the random variables Xi have the Weibull distribution (4.3) t h e n , as t+CO,

The last statement in (6.5) (for 0 < X < X I ) readily follows from Theorem 5.1 for 0 < a < 1 [see also (6.6) and (6.7) below] using the fact that B ( t ) /E[s~(t)] -+ 0 as t + 00. Indeed, note that

and, by the Scaling Assumption (5.1) and regularity of the functions H ( t ) Ho(t) with index p’,

-

where the last inequality follows, by the substitution (2.10), from the elementary inequality [see Hardy, Littlewood and P6lya (1952), Section 2.15, Theorem 41, page 391

1 - =e‘

> e/ae‘-l (1 - a )

(0 < a

< 1, p’ > 1).

Analogously, Theorems 3.2 and 5.4 yield the following united assertion:

If the random variables Xi have the Weibu11 distribution (4.3) t h e n , as t h o ,

-{

E[Sjv(t)] (Var[SN(t)l) 1’2

sN(t)

-

d

N(o,1) if N(0,

if

> A 2 ( a > 21,

x = A2

( a = 2).

Finally, Theorem 5.1 takes the following form: If X i have the Weibull distribution (4.3) t h e n , as t -+ 00,

where the stable law 3, is described in Theorem 5.1 and

E[s~(t)] if X i < X < A2 (1 < if if with B l ( t ) given by (6.3).

x = A1 < x < A1

0

A(t) is of the form < 2),

( a = 11, ( 0 < a < l),

(6.7)

61

7. Sketch of the proofs Theorems 5.1, 5.3 and 5.4 can be proved using the known methods for sums of independent random variables [see Gnedenko and Kolmogorov (1968) and Petrov (1975)]. However, the actual proofs are technically quite involved, because we have imposed only very minimal smoothness conditions on the distribution of X . So the full details are not given here, but rather will be published elsewhere. Nevertheless, it is not difficult t o explain the main points behind the calculations. In particular, it is important to clarify the central role and power of the Basic Identity (4.11). The key step in the proofs is the evaluation of the tail probability [cf. Petrov (1975), Chapter IV, 5 1, 21 P{etx

> x B ( t ) )= P { e t x > xexp[p(t)~o(t)l}

=

pix > 17z(t)}

= “xp[-h(17z(t))]1

where we used the notations (2.1) and (4.7). This expression needs to be compared t o the sample size, N eXHo(t),and therefore we have t o relate the function h ( q z ( t ) )to the canonical scale determined by the rate function Ho(t). In so doing, equation (4.11) plays the major role, as well as the following lemma. N

Lemma 7.1. For each x > 0, t-+m lim

Proof. Note that, as t

-+

[ h ( v z ( t )-) h(771(t))]= culogx. 00,

% ( t )- 171(t)

=

log x t

-+ o

and

By Taylor’s formula and normalized regular variation of the function h [see (4.1)] we have, as t + 00,

62

Using the Basic Identity (4.11), the right-hand side can be rewritten as

ex -1ogz At)

+ alogx

(t -4 oo),

according t o (4.10), and the lemma is proved.

0

Let us now obtain the main ingredient of the limiting infinitely divisible law -the Lkvy-Khinchin spectral function C,using the formula

[see Petrov (1975), Chapter IV, 5 2, Theorem 8, pages 81-82]. First of all, note that C ( x ) = 0 if x < 0. For z > 0, using the Scaling Assumption (5.1) and formula (7.1) we have

-

NP{etx > z B ( t ) } exp{XHo(t) - h ( q z ( t ) ) } .

(7.3)

The Basic Identity (4.11) and Lemma 7.1 imply XHo(t) - h (v z(t ))= h(v1(t))- h ( v z ( t ) ) + -crlogz (t + m).

(7.4)

Hence, returning to (7.2) we obtain -C(x) = >i&exp{AHo(t) - h ( v z ( t ) ) } = exp{ -a log x} = 2-0,

and therefore a is indeed the characteristic exponent of the limiting law. 8. Limit distribution of the maximum Consider the partial maximum

MN(t) :=max{etxi, i = 1, . . . , N } = e x p ( t X I , ~ ) , where

X~,N := max{Xi, i = 1,. . . , N } . Recall the notation (5.2),

B ( t ) = exP{P(t)Ho(t)}.

63

Theorem 8.1. For all X

> 0 , as t -+ 03,

where i9, is the Fre'chet distribution, with distribution function exp(-z-,)

zf

@n(x)=

z

> 0,

otherwise.

Proof. As before [cf. (7.1)], for z > 0 we have p{MN

5 z B ( t ) }= P { X l , N 5 % ( t ) }

-+

exp(-z-")

(t

-+

00),

0

as shown in (7.3) and (7.4).

Remark 8.2. The Frkchet distribution a, is one of the three types of possible weak limits for maxima of i.i.d. random variables [see Galambos (1978), Sections 2.1 and 2.41. However, the known general theorems about convergence of the maximum to i9, are not directly applicable in our case. From Theorem 8.1, it is easy t o derive a logarithmic Law of Large Numbers for the maximum. Theorem 8.3. For all X

> 0 , as t

4 00,

logMN(t) 3 HO ( t )

($)"'

Proof. Taking the logarithm of M N ( t ) and dividing by Ho(t) -+ 00, from Theorem 8.1 we deduce that

whence our claim follows.

0

It is interesting t o compare the maximal term M N ( t ) with the entire sum SN(t). In fact, Theorem 5.1 implies the following Law of Large Numbers for logSN(t), which can be seen as providing an analogue of the limiting free energy F ( P ) in the Random Energy Model [see (1.4)].

64

Theorem 8.4. For all X > 0, as t

a,

Comparing Theorems 8.3 and 8.4, we note that in the case 0 < X 5

A1

which indicates that the contribution of the maximal term MN t o the sum SN is logarithmically equivalent t o the whole sum. In the opposite case where X > X I , the limit in (8.1) is strictly less than 1, so that the maximum MN(t) is negligible as compared to the sum SN(t). This observation is supported by the LLN being valid for X 2 A 1 (see Theorems 3.1 and 5.3).

Acknowledgments The authors wish t o thank Anton Bovier for useful discussions. Several visits to the EPFL (Lausanne) and Universitat Bonn during this research have been very stimulating, and the hospitality of these institutions is much appreciated. The second author gratefully acknowledges travel grants from The Royal Society and the University of Leeds, which made possible his participation in the First Sino-German Meeting on Stochastic Analysis (Beijing, 2002) where the results of this paper were presented. Support from the local organizers is also appreciated.

References [l] ATHREYA,K . B. AND NEY, P. E . (1972). Branching Processes.

Springer, Berlin. [2] VON BAHR,B. AND ESSEEN, C.-G. (1965). Inequalities for the r t h absolute moment of a sum of random variables, 1 5 r 5 2. Ann. Math. Statist. 36, 299-303. N. H., GOLDIE,C. M. A N D TEUGELS,J . L. (1989). Reg[3] BINGHAM, ular Variation. Paperback edition (with additions). Cambridge Univ. Press, Cambridge. [4]BOVIER,A . , KURKOVA, I. AND LOWE, M. (2002). Fluctuations of the free energy in the REM and the pspin SK model. Ann. Probab. 30,605-651. [5] DERRIDA,B. (1980). Random-energy model: Limit of a family of disordered models. Phys. Rev. Lett. 45, 79-82.

65

[6] EISELE,TH.(1983). On a third-order phase transition. Cornm. Math. Phys. 90, 125-159. [7] GALAMBOS, J. (1978). The Asymptotic Theory of Extreme Order Statistics. Wiley, New York. [8] GNEDENKO, B. V. AND KOLMOGOROV, A. N . (1968). Limit Distributions for Sums of Independent Random Variables. 2nd ed. AddisonWesley, Reading, Mass. [9] GRADSHTEYN, I. S. A N D RYZHIK,I. M. (1994). Table of Integrals, Series, and Products. 5th ed. (A. Jeffrey, ed.) Academic Press, Boston. [lo] HARDY,G . H., LITTLEWOOD, J . E.AND P 6 L Y A , G . (1952). Inequalities. 2nd ed. At the University Press, Cambridge. [ll] IBRAGIMOV, I. A. AND LINNIK,Y u . V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen. [12] OLIVIERI,E.AND P I C C O , P . (1984). On the existence of thermodynamics for the Random Energy Model. Comm. Math. Phys. 96, 125-144. [13] PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin. I141 PETROV, V. V . (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford. [15] RESNICK, S . I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. [16] SCHLATHER, M. (2001). Limit distributions of norms of vectors of positive i.i.d. random variables. Ann. Probab. 29, 862-881.

Thin-Film- G rowth-Mo dels : On local solutions Dirk Blomker Mathematics Research Centre, University of Warwick, Coventry CV4 7AL - UK [email protected] Christoph Gugg Lehrstuhl fur Analysis und Modellierung, Fachbereich Mathematik, Universitat Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart , Germany [email protected]

Abstract Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of unique solutions. We obtain a gap of regularity between the unique local and the global solutions, which are necessary to define statistical quantities like mean interface width or correlation functions.

Keywords. martingale solution, existence, local solutions AMS subject classification. 60H15

1. Introduction In this article we consider the existence of solutions for nonlinear stochastic partial differential equations (SPDEs) arising in thin-film-growth. Global existence of solutions in space-dimension one is proved via a Galerkin method (see [5]), but this solution does not exhibit very high regularity. This paper focuses on the local existence of unique solutions. We will see that there remains a huge gap between the regularity of the local solutions and the global solution. It remains open whether the global solution is unique or not. 66

67

In the models under consideration the temporal evolution of a surface is described by its height profile h ( t , z ) at time t 2 0 over the space point z E G c Rd. We treat recently proposed models for thin-film-growth (see [23]),where the evolution of the height profile h is given by a stochastic partial differential equation of the form

&h = Ah

+ ClVh12+ [

for t

>0,

h(0) = ho.

(1.1)

Here A and C denote linear differential operators and ho is some random initial condition. The additive stochastic noise term is given by a generalized stochastic process [ = { [ ( t z) , : t 2 0, z c G} which is the generalized derivative of a certain Wiener process in a Hilbert space. The main example we are interested in arises in the theory of thin-filmgrowth models (see e.g. [23]). It is given by:

+

L$h = -alA2h - ~ 2 A h (QA - a4)[Vhl2 [

(1.2)

The coefficients ai are usually positive constants, but it is possible to relax this constraint for i > 1. However, for global existence (see [5]) it is essential that the differential operator (a,A - a 4 ) is either positive or negative. The above equation is considered on [0,Lid, d = 1 , 2 subject to periodic boundary conditions and with deterministic initial value ho = 0 corresponding to an initially flat surface. The case d = 2 corresponds to the full three-dimensional growth model and is therefore of particular interest. For the physical derivation of (1.2) or the related equation with a4 = 0 we refer the reader to Siegert, Plischke [24],Raible, Linz, Hanggi [21, 22, 161, or [23]. A general survey on surface growth processes and molecular beam epitaxy is given in Krug and Spohn [15], Meakin [18],Barabasi and Stanley [3], Halpin-Healy and Zhang [12], or Krug [14],just to name a few. Another related model that fits in the general framework of (1.1) is the Kuramoto-Shivashinski equation where a3 = 0. The existence and uniqueness of global solutions in case d = 1 is established by Duan and Ervin [7]. But it is still open for general two-dimensional domains. The existence of global solutions in the case d = 1 for (1.2) has been addressed recently in [4] and [5]. There the authors verified the existence of a bounded L2-valued solution, but could not show the uniqueness of the solution. In [6] the existence of stationary solutions was shown for a subclass of equations. To the best of our knowledge, there are no results availa.ble on global solutions in dimension d 2 2, mainly because it is not known how to derive a-priori estimates in the presence of the quadratic terms. Nevertheless it

68

should be possible to obtain global solutions on small strip-like domains in R2, as this was done e.g. for the Kuramoto-Shivashinski equation. The main aim of this paper is to provide a proof of unique local solutions for (1.1) by standard fixed-point arguments. As the global solution is not very regular, we try to use spaces of functions with as less regularity as possible. Unfortunately there remains a gap, and it was therefore not possible to verify the uniqueness of the global solution. Nevertheless local solutions are not useful, if we want t o obtain finite expectations of the solution. Moreover, we cannot define statistical quantities like the mean surface roughness or mean correlation functions, which play an important role in the physical literature. See for example [23], where unknown coefficients are determined from real world and numerical experiments. In [5] the numerical convergence of a spectral Galerkin scheme for such quantities was investigated. The paper is organized as follows. In Section 2 we introduce some basic notation and state all assumptions needed in the paper. Section 3 contains the result for the existence of a global solution. We establish the existence of local solutions in Sections 3 and 4, first in the Sobolev space H1(G), which applies only to dimension d = 1, and later in the space W1l4(G), which is useful for d = 1 , 2 , and 3.

2. Assumptions In this section we summarize a few assumptions valid throughout the whole article. Suppose A is a fourth order linear differential operator (e.g. -alA2 a2A with periodic b.c.) subject to suitable boundary conditions in L2 = L2(G) for some domain G c Rd with sufficiently smooth boundary. Moreover, assume that A generates a strongly continuous analytic semigroup {etA},>o - in L2. Now define H4 = D ( A ) , which is the subset of functions from the Sobolev space H4(G) fulfilling the boundary conditions corresponding to A . Moreover define the fractional Sobolev spaces H" for a 2 0 by H a = D ( ( w 1 - A)"/'), where w is a sufficiently large constant. We equip H" with the standard graph norm. The dual space of H" is denoted by

H-". It is now easy to verify (see e.g. Section 2.6 of [20]) that for any y 2 p and any T > 0 there is a constant M > 0 such that IletAuII~-, 5 M t - v l l u l l ~ o for t E (O,T].

(2.1)

69

Our main example for C is given by Equation (1.2). Here C = -a3A+a4. In general we only assume that C is a continuous operator from H 2 to L2, which can be extended to a continuous operator from H2+’ to H 6 for any b E R. To reformulate the problem as an integral equation we need the generalized stochastic process { J ( t ) } t 2 0 to be the generalized derivative of some Q-Wiener process W = {W(t)}t>o - on a probability space (52, A, P ) with values in some Hilbert space. Moreover, W is adapted to some filtration {Ft}t?o. As usual, Q denotes the covariance operator of W(1). Moreover, we suppose that P-almost sure W has values in C([O,T], H”O), for some SO E R, and that the stochastic convolution t

WA(t) =

e(t-T)AdW(T), t 2 o

is well-defined. The space in which we actually consider WA will be the topic of further assumptions. Usually, we do not have enough regularity of W or WAto solve (1.1), so we only obtain solutions in some generalized sense. Therefore, we consider mild solutions, defined as Hilbert- or Banach-space-valued solutions of the corresponding integral equation, called the variation of constants formula,

where we define B ( u )= IVuI2. To prove the existence of solutions for (2.2) we first define v Hence,

=

h - WA.

In general v is much more regular than h, for example it is weakly differentiable in time. Note that the transformation from (2.2) to (2.3) is not stationary, as in many articles (see e.g. [13] and the references therein). This is not necessary in our case, as we only consider existence of local solutions. The following equality is helpful to extend a local solution. Let TI and 7 2 be positive stopping times, then

70

3. Global solutions In this section we recall the results of [5]. The problem is now to construct a solution h : [O,T]x R 4 L 2 ( [ 0 , L ] )(i.e., G = [O,L] and d = 1) of the equation

+

+

dh(t) = Ah(t)dt LCB(h(t))dt dW(t),

h(0) = ho

(3.1)

for t E [O,T]with arbitrary T > 0. A and L are now the operators given in Equation (1.2) subject to periodic boundary conditions. Moreover ho is some random initial condition ho E L 2 ( [ 0 , L ] fulfilling ) the following assumption.

Assumption 3.1. Let E[1nf(IlhollL2)]" < 00 for some K: 2 1, with lnf(z) = max(0, In(.)}. For the stochastic convolution we assume that the following assumption is valid. On the one hand it allows to apply a spectral Galerkin-method in a simple way, on the other hand it is a requirement on the regularity of the stochastic convolution.

Assumption 3.2. Suppose that Q and A have a joint complete orthonormal system of eigenfunctions. Moreover, we assume tr(Q(1 - A)'-') < CQ for some 6 > 0. We cannot prove the existence of a mild solution

of Equation (3.1) on

(R, A, P ) that satisfies Equation (2.3), but we will change the underlying probability space to construct a martingale solution (cf. [8], Section 8). A stochastic process h defined on a different probability space M = (h,2,p ) is called a martingale solution of (2.2), if the corresponding 6 =

h - w ~satisfies (2.3) with KO as initial condition. Here, KOis defined on M , and W Ais defined by a Q-Wiener process W on M such that the common distribution of KO, W , and @A equals the distribution of ho, W , and W A . In [5] we used the Galerkin-method to prove the existence of V or h. By P(") we will denote the L2-orthogonal projection onto the span of the first 2n + 1 eigenfunctions of Ape*. Define W y ' ( t ):= P'"'WA(t) and h g ) := P(")ho,and suppose that d")is the solution of

The existence and uniqueness of a solution dn)of the stochastic ordinary differential equation is standard (cf. [2], Section 6.3 or [19]). The main technical result is the following a-priori bound (see [5]).

71

Lemma 3.3. Suppose d = 1 and C positive. For some n 2 1 and any T > 0 let Assumption 3.1 and 3.2 be true. Then there is a positive constant CT independent of n E N,such that

(3.3)

In [5] it was possible t o establish the weak convergence of a subsequence of the probability distributions P"'") , n E N,by compactness arguments. We obtained the following main result:

Theorem 3.4. Consider equation (1.2) for d = 1 with positive operator C subject t o random initial condition ho E L 2 ( [ 0 , L ] )Fix . L > 0 and T > 0 and let Assumptions 3.1 and 3.2 be true. Then there exists a martingale solution h on a probability space d,P ) such that

(a,

I

=h-

I

WAE L 2 ( [ 0 i T I i H ~ e r ( [ ~ i~LCI )()[ O , T I , H , - , ~ , ( [ ~ , L I ) )

P -almost sure and ij satisfies (2.3) with r?l~and ho instead of WA and ho in the space L 2 ( [ 0 , T ] , H ~ , , ( [ 0 , L ]P) ) -almost , sure. Remark 3.5. The previous theorem does not state the optimal regularity for our solution. For a detailed discussion see [4]. The best we can hope for is ij E L " ( [ 0 , T ] , L 2 ( [ 0 , L ]n)L) 2 ( [ o , T ] , H ~ e , ( [ o , Ldue ] ) )t o the a-priori estimate. Using an interpolation inequality this immediately implies ij E L4([07T17H j e r ( [ O i ~51)). Moreover, we can improve the continuity results a lot, but it is still an open problem whether there is a continuous solution with values in L 2 ( [ 0LI). , Remark 3.6. The result of Theorem 3.4 remains true if we extend the time interval to the whole positive time axis R$. In this case we obtain a global solution V in L~o,(R$,Hi,,([O, L ] ) ) n C ( R $H;t([O, , 4)).For a detailed discussion of a similar problem see [6] for the existence of stationary solutions. 4. Local existence in H' In this section we focus again on d = 1 and G = [0,L ] ,but we admit general boundary conditions as Dirichlet or Neumann b.c., e.g.. At least, the b.c. must be such that the operator A fulfills the assumptions of Section 2.

72

For the proof of local existence, we cannot use the nice Gelfand triple

H-2 c L2 c H 2 suggested by the a-priori estimate for periodic boundary conditions. First the nonlinearity is not defined for L2-valued solutions. Moreover, even for u E Lm([0,TI, L2) n L 2 ( [ 0TI, , H 2 ) we obtain in general LB(u) L 2 ( [ 0 , T ] , H - 2 )which , is essential in many proofs (see e.g. [9] for the stochastic Navier-Stokes equation). Define the mapping 6 by

[G(u)](t) :=

1

LCe(t-+u(s)ds.

We verify the following:

Proposition 4.1. For -2 I y-P < 2 - 8 and a n y p E (1,co)the mapping 9 : LP( [0,TI,220) 4 C ([0,T ],H Y ) is a continuous linear operator. Moreover,

IIG(u> I I L z p ( [ O , T ] , H ~5) CTIIuIILp([o,T],H@) where CT is a generic constant with CT -+0 for T -+ 0.

(4.1)

7

Remark 4.2. Even compactness holds for the embedding result in the previous proposition. For p = 1 and y - p = -2 the results are still valid, but the embedding will no longer be compact (see for example [lo]). Proof: For the proof of inequality (4.1) note that by (2.1) and Holder inequality

This immediately implies (4.1). Moreover, we immediately see that 6 : Lp([O,T], HP) + L"([O,T],H Y ) is a continuous linear operator. The fact that G actually maps into the space of continuous functions is straightforward. 0 With the help of the previous proposition we now consider the Lipschitz continuity of G(B(.)).We employ the Sobolev embedding from L1 into HB for p < -d/2. Therefore we impose y E (-2 - $, 2 - $ = -

8)

(-s,

9

73

4) which allows to choose some ,B < -1/2 such that the requirement of P Proposition 4.1 is fulfilled. (Note that for d = 2 and y 2 1 this condition is empty.)

IIG(B(u)) - G ( B (w)) [I%'( [O,T], H Y )

I 'TIIB(.)

-

B(W)ll$([o,T],HP)

where CT again denotes different constants with C,

IIG(B(u))

-

+0

for T

provided p

Moreover it is straightforward to verify that B : H1 ,B < -1/21 and, additionally] B is locally Lipschitz with -

0. Hence,

G(B(W)>ll L 2 p ( [O,T],H1)

Ic~lb + w11L2P((O,T],H1)IIu -WIIL~P([O,TI,HI)

lB(.)

4

WW)IlHP

4

> 8.

HP for any

I Cll(&. + &W)(%. - @ZW)IILI I Cll. + W l l H l .1 - W l l H l .

From the proof of Proposition 4.1, we immediately obtain for -1+;

11 G(G)11 C ([O,T],H1) 5 cT I I I I C ([O,T],HO) .

-;> /3 > (4.2)

Hence, G(B(.))is locally Lipschitz in C([O,T],H1) with

llG(B(u)) - G(B(W))IIC([O,T],H1) I cTllu Theorem 4.3. Fix p > 8 and d = 1 and

f W I I C ( [ O, T ] , H 1 ) I I u wIIC([O,T],H1).

assume that W, E C(R,f,H1)Pa.s. (which is true e.g., if Q is a cont. linear operator in L2 commuting with A ) . Suppose that ho is some random initial condition with lletAhOllH1 E L2P([0,T]) (this is fulfilled if ho E H s for 6 > 1Then there is a unique

g).

mild solution v E L12,",([0,7*),H1) n C((O17*),H1) of (2.3), where T * is a P - a s . positive stopping time. Moreover] it is possible to choose T * such that the following is true with probability one: Either T * = 03, or otherwise Ilv(t)ll~i-+ 03 for t -+ T * and J;* Ilv(s)ll$lds = co.

74

It is even possible to consider the asymptotics of Ilv(t)llH1 for t 0 (cf. Section 7 of [17]). Nevertheless, we do not obtain sufficiently less regularity of our unique solution to verify the uniqueness of the global solution of the previous section, which was either in L4([0,T],H1)or in L"([0,T],L2). The global solution constructed in the previous section, will have nonuniqueness if the H1-norm blows up. But we know that a blow-up has to be fast enough, as the @-norm fails t o be in L2P([0,T*]) for p > 8. It is possible to describe this blow up in more detail, but we will not focus on that. Proof (Theorem 4.3): We just give a brief sketch how to construct the solution path-wise. Define the operator 'T by [I(.)]@) := etAho [6(B(u WA)](t). Standard fixed point arguments for I imply the existence of a fixed point w E L 2 P ( [0, 71, H1),where we have to choose T sufficiently small. Using Proposition 4.1 and the regularity of the semigroup, it is easy t o verify that we already have 21 E C((O,.r],H1).Now we can use the local Lipschitz continuity of I in C ([0,TI, H1) together with fixed point arguments for the representation in (2.4) t o extend the solution until T * = 00 or ((v(T+)((Hl = 00. Note that T * := sup{t > 0 : ((v(t)lJH1< a}. Moreover we easily see by (2.3) that w is adapted t o the filtration of the Wiener process W . If T * < 00, we estimate for T 5 t < T * --f

+

+

5. Local existence in W1y4 The existence result of the previous section is only valid for dimension d = 1. We have to employ spaces of smoother functions t o obtain a result for d > 1. In order to use some results of the previous section we restrict ourselves for simplicity to W174, which is the Sobolev space of functions with first derivatives in L4. This will apply to d = 1 , 2 , 3 . To treat higher dimensions, we could use W1>qspaces with q large. Moreover it should be possible to reduce the 4 a little bit. In both cases we only need LP-estimates similar to (2.1), which are well-known. Using the Sobolev embedding of H1+%into W1l4,we obtain by Propo-

75

sition 4.1 IIG(u)IIL2p([0,T],W1,4)5 CIIuIILp([0,T],L2)

and

I16).( IIC( [O,TI,W 1*4)5 II II

L p ([O,T],

Lz)

+

provided -2 5 1 < 2 - 4/p. Hence, p > 16/(4 - d). Obviously by the Cauchy-Schwarz inequality

5 ClIu + wllw1,4\\'1L- 'uIIIw1,4. This immediately implies

llG(B(u>)- Q(B(~))IIL2p([o,T],W1,4)

5 cTllu

+ 'WIIL2p([0,T],W1,4)11u- w ( I L z ~ ( [ o , T ] , W 1 ~ 4 ) .

Moreover, we obtain that G(B(.))is also locally Lipschitz in C([O,TI,W1>4) with

llG(B(u>> -~(B(~))llc(Io,T],w~~4)

5 CTllu -k wIIC([0,T],W114)/ I u

- wIIC([0,T],W1*4).

Now we can prove the following theorem completely analogously t o Theorem 4.3.

Theorem 5.1. Fix d E {1,2,3} and p > 16/(4 - d), and assume that W A E C([0,T],W1y4). Suppose that ho is some random initial condition with IletAh011wi,4 E L2P([0,T]).Then there is a unique mild solution w E Lfrc([O,~*), W1l4) n C((O,-r*),W1y4) of (2.2), where T * is a P-a.s. positive stopping time. Moreover, it is possible t o choose T* such that the following is true with probability one: Either T* = 00, or otherwise we obtain I l w ( t ) l l ~ i , 4 + 00 for t 4 T * and ~ / ~ ( . s )= ~ 00. ~ ~ ~ , ~ d s

Jl*

We remark without proof that the regularity WA E C((0,TI, W134) required for the previous theorem is in general not guaranteed for space-time white noise in dimension d > 1, which is the noise necessary for the applications. Note that here the covariance operator Q is the identity operator. For d > 1 we need smoother noise. Consider our example in (1.2). On the interval there is a version of WA which is not only in C([O,T], H:,,([O,T])) but already in

76

C([O,T],C~,,([O,T])),as was e.g. verified in [ l l ] . But this proof relies heavily on Lm-bounds for the eigenfunctions of the differential operator A , which are very restrictive for d > 1. References [l]R . A. ADAMS.Sobolev Spaces, volume 65 of Pure and Applied Mathematics. Academic Press, Inc., 1978. [2] L. ARNOLD.Stochastic Differential Equations: Theory and Applications. John Wiley and Sons, 1974. [3] A. L. BARABASI A N D H. E. STANLEY.Fractal Concepts in Surface Growth. Cambridge University Press, 1995. [4] D. BLOMKERA N D C . GUGG.On the existence of solutions for amorphous molecular beam epitaxy. Journal of Nonlinear Analysis: Series B Real World Applications, 3(1):61-73, 2002. [5] D. BLOMKER,C . GUGG, A N D M. RAIBLE. Thin-Film-GrowthModels: Roughness and Correlation Functions. European Journal of Applied Math., 13:385-402, 2002. [6] D. BLOMKER AND M. HAIRER. Stationary solutions for a model of amorphous thin film growth. Stochastic Analysis and Applications, to appear. [7] J . DUANAND V. ERVIN. On the stochastic Kuramoto-Sivashinsky equation. Nonlinear Analysis, 44:205-216, 2001. [8] G . DA PRATO A N D J . ZABCZYK. Stochastic Equations in Infinite Dimensions. Number 44 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1992. [9] G . DA PRATO AND J . ZABCZYK. Ergodicity for infinite dimensional systems.. London Mathematical Society Lecture Note Series. 229. Cambridge Univ. Press, 1996. [lo] D. GATAREK.A note on nonlinear stochastic equations in Hilbert spaces. Statistics&Probability Letters, 17:387-394, 1993. [ll] C . GUGG, H. KIELHOFER,A N D M. NIGGEMANNOn the approximation of the stochastic Burgers equation. Commun. Math. Phys. 230(1):181-199, 2002. [12] T. HALPIN-HEALY AND Y . C . ZHANG.Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Physics Reports, 254:215-414, 1995. [13] P. IMKELLER A N D B. SCHMALFUSS The conjugacy of stochastic and random differential equations and the existence of global attractors. J.

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Dyn. Differ. Equations 13(2):215-249, 2001. [14] J. KRUG. Origins of scale invariance in growth processes. Advances in Physics, 46:139-282, 1997. [15] J. KRUGA N D H. SPOHN. Kinetic roughening of growing surfaces. In C. Godrhche, editor, Solids f a r f r o m Equilibrium, pages 479-582. Cambridge University Press, 1991. [16] S. J. LINZ, M. RAIBLE, AND P. HANGGI. Stochastic field equation for amorphous surface growth. volume 557 of Lecture Notes in Physics, pages 473-483, 2000. [17] A. LUNARDI. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhauser, 1995. [18] P. MEAKIN.The growth of rough surfaces and interfaces. Physics Reports, 235:189-289, 1993. [19] B. 0KSENDAHL. Stochastic Differential Equations. An I n t r o d u c t i o n with Applications. Springer, 1995. 4th edition. [20] A. PAZY. Semigroups of Linear Operators and Application t o Partial Differential Equations. Number 44 in Applied Mathematical Sciences. Springer, 1983. [21] M. RAIBLE,S . J. LINZ, AND P. HANGGI. Amorphous thin film growth: Minimal deposition equation. Physical Review E, 62:16911705, 2000. [22] M. RAIBLE,s. J. LINZ, AND P. HANGGI. Amorphous thin film growth: Effects of density inhomogeneities. Physical Review El 64:31506, 2001. [23] M. RAIBLE, S. G. MAYR,S. J. LINZ,M. MOSKE,P. HANGGI,AND K . SAMWER.Amorphous thin film growth: Theory compared with experiment. Europhysics Letters, 50:61-67, 2000. [24] M. SIEGERT AND M. PLISCHKE. Solid-on-solid models of molecularbeam epitaxy. Physical Review E l 50:917-931, 1994.

Asymptotic behaviour of solutions to the 2D stochastic Navier-Stokes equations in unbounded domains new developments Zdzislaw Brzeiniak and Yu-Hong Li Department of Mathematics, The University of Hull, Hull HU6 7RX, U.K. email: [email protected] and [email protected]

Abstract This paper is concerned with the large time behaviour of solutions to the 2D stochastic Navier-Stokes equation on some unbounded domains. A notion of asymptotic compactness of a Random Dynamical System on a Polish space, introduced by authors in [ll],is recalled. We present some results from that paper as well as announce some results from [12]. Firstly, we show that an asymptotically compact RDS on a separable Banach space has a non-empty compact 0-limit set. Secondly, we show that the RDS generated by a stochastic NSEs on a 2D domain satisfying the Poincarb inequality is asymptotically compact on the space H of divergence free vector fields on D with finite L2 norm. Thirdly, we show that this is also true for the space V of divergence free vector fields on D with finite norm. The first two results have been proved in [ll],the third comes from the work in preparation [12]. This research is motivated by papers [35] by Rosa and I311 by Ju, where similar questions for dynamical systems generated by 2D deterministic NSEs were studied. One consequence of our main results is the existence of a Feller invariant measures (however the uniqueness of invariant measures remains an open problem). The case of stochastic NSEs in bounded domains have been studied earlier in [4], [8] and [38].

1991 Mathematics Subject Classification. Primary 60H15, 60H30; Secondary 37H10,76DO6 78

79

Keywords and Phrases. Random dynamical systems, Asymptotic compactness, R-limit set, Weak continuity, Random absorbing set, Stochastic Navier-Stokes equation, Poincarb inequality

1. Introduction To better understand the large time behaviour of solutions to some Stochastic Partial Differential Equations (SPDEs), the qualitative theory of such equations plays an important r61e in their mathematical study. As was pointed out in [2] by Ludwig Arnold, around the year 1980 Elworthy, Baxendale, Bismut, Ikeda, Watanabe and others have discovered that a stochastic differential equation generates a flow of random diffeomorphisms which has a much richer structure than simply a family of stochastic processes and solutions of the stochastic differential equation for a given initial value. So one can bridge the gap between stochastic analysis and random dynamical system by proving that random or stochastic differential equations generate a random dynamical system. Once such a random dynamical system is constructed, its properties become an interesting problem. For example, one may study Lyapunov exponents, see e.g. Elworthy-Carverhill [16], Arnold [2] and references therein. The reason for this is that a general result of Ruelle [37] on the existence of Lyapunov exponents and Lyapunov manifolds is then applicable. For stochastic PDEs the problem of the existence of RDS is much more subtle. In fact there is so far no satisfactory theory of stochastic flows for stochastic PDEs. First results in this direction were obtained by Flandoli and Schaumloffel [29] who proved the existence of RDS for some linear parabolic SPDEs and showed that the Ruelle theory was applicable. Later this question was pursued by many authors, see e.g. [9]. The case of nonlinear problems turned out to be even more difficult. The existence of a (local) RDS for 2D stochastic NSEs was first proved in [7], see also a paper [15] by Capiriski and Cutland. However, the problem of the existence of an attractor was not resolved until the works by Crauel and Flandoli [21] and independently by Schmalfuss [38] (see also an earlier work [8] for the case of real noise). The difficulty was to find a proper definition of an attractor and this difficulty was resolved in the above quoted papers. Applications to 2D NSEs in bounded smooth domains were also given in these papers. In recent years, there has been a growing interest in the problem of the existence and uniqueness of an invariant measure, see e.g. Flandoli and G3tarek [27], Flandoli and Maslowski [28], Ferrario [26], Bricmont [3],

80

Kuksin and Shirikyan [32] and E, Mattingly and Sinai [24]. But all these works on large time behavior of the solutions for 2-D stochastic NavierStokes equations are based on the compactness of the imbedding V v H , which is valid for bounded domains (see e.g. [27], [28], [26], [3], [32], [24] [39], [20], [21], [4], [5] and some others). For unbounded domains the compactness of the above imbedding does not hold any more. We need to use a new method to overcome this difficulty. As we showed in our previous paper [ll],a new concept of asymptotically compact random dynamical systems is the method one has been looking for. The current paper is a kind of amalgamation of our two papers [ll] and [12]. We present the results from both of them, providing a proof for only a portion of them. Let us now describe briefly the content of the current review paper. In section 2 we recall the definitions related to RDS. In particular, we give the definition of the crucial new notion of asymptotic compactness. In section 3 we study the initial value problem for the stochastic Navier-Stokes equations. As assumed in many recent papers, see e.g. [41, ~51,[31, [ ~ o I[211, , ~ 4 1 ~, 6 1 ~, 7 1 PI, , ~ 2 1 ~391 , and therein, our noise is additive. We seek a solution u in the form v z , where z is the corresponding stationary Ornstein-Uhlenbeck process. We assume that such a process with certain regularity properties exists, see Assumptions A.3 and A.4. Since we formulate our results for both spaces H and V , the assumptions above correspond to these two cases. In contrast to our previous work [ll]we do not use Galerkin approximation to prove the existence but instead, motivated by [4], we apply the Banach fixed point theorem in appropriately chosen spaces to prove the local existence and then employ a’priori bounds to infer the global existence. Our results imply the existence of RDS generated by our SNSEs both in the space H and V , see Theorem 3.21. In section 4, we prove that for the RDS generated by our SNSEs, every bounded set B possesses a random closed and bounded absorbing set K which depends on B , and this does not imply the existence of an attractor. This is stated and proved in both cases, H and V . In section 5 we prove the technical but important result that our RDS is weakly sequentially continuous in certain appropriate topologies. Section 6 is devoted to proving that our RDS on V is asymptotically compact. A corresponding result from [ll] on asymptotic compactness on H is used. Finally in section 7 we briefly discuss the existence of a Feller invariant measure. This result can be viewed as an application of our main results. As mentioned before, uniqueness of the invariant measure remains an open problem. We will come back to it in our later work.

+

81

2. Preliminaries on RDS's

A basic notion in this section is of a metric dynamical system (DS), 1.e. ' a quadruple % = P,37P,d),

where (R, 3 , P ) is a probability space and 6 = (6),,wsatisfies the following four conditions: 6 : BxR 3 ( t ,w)w 6,w E R is a (B(R)@3,3)-measurable; for each t E B, 6t : R 3 w H 6 ( t , w ) E R preserves IF'; for all t , s E R &+, = Ot o 6, and 60 = id. The following definition is fundamental, see [2] for details.

Definition 2.1. Assume that Z = (R,F,P,6)is a metric DS. Suppose that ( X , d ) is Polish space and B = B ( X ) is the Borel c-field on X . A map cp : R+ x R x X + X,( t , ~s ), H p(t,w)x, is called a measurable random dynamical system (RDS), iff (i) cp is (B(W+) @ 3@I B,B)-measurable. (ii) cp(t s , w ) = cp(t,6,w) 0 cp(s,w) for all s , t E It+, and cp(0,w) = id for all w E R.

+

Remark 2.1. Property (ii) above is usually called the (cocycle property). One can also define RDS with R+ replace by we are only concerning with the case R+.

R,Z or N,but

in this paper

A RDS 'p is said to be continuous iff P-as. for all t E R+, 'p(t,w ) : X -+ X is continuous. Similarly, a RDS is said to be time continuous iff P-a.s. for all x E X, p(.,w)x : R+ -+ X is continuous. Let, for A , B c X , d ( A , B ) = supZEAd(x,B)and p(A,B) = max{d(A, B ) ,d(B,A ) } . The latter is called a Hausdorff distance. Then the the class C of all closed subsets of X with the Hausdorff distance become a metric space (see Castaing and Valadier [17]). Let us denote by B(C) the class of all Borel subset of C (with respect the Hausdorff metric). We have the following:

Definition 2.2. A set valued map C : R -+ C is said to be measurable (or a closed random set) iff C is (F,B(C))-measurable. A random set C will usually be denoted by C ( w ) . Definition 2.3. A closed random set K ( w ) is said to absorb a random set B ( w ) iff IF'-as. there exists a time t ~ ( w ) called , the absorption time, such that for all t 2 t B ( w ) , cp(t,6-tw)B(6-tw) C K(w).

a2

Definition 2.4. A random set B is said to be p-forward invariant iff y ( t , w ) B ( w )C B ( & w )

for

all

t > 0.

(2.1)

A random set B is said to be strictly 9-forward invariant iff cp(t,w)B(w)= B ( & w )

for

all

t > 0.

(2.2)

Remark 2.2. Substituting 6-tw for w, we have the following equivalent version of Definition 2.4. A random set B is p-forward invariant, resp. strictly pforward invariant, iff

respectively

‘p(t,d-tw)B(d-tw) B ( w ) for all ‘p(t,O-,w)B(6-,w) = B ( w ) for all

t > 0, t > 0.

Definition 2.5. Given a random set B , the S2-limit set of B is the set

nu

s ~ ( B= , ~~ ) 2 ~ =( ~ )

cp(t,d-tw)q8-tw).

(2.3)

T20 t2T

Remark 2.3. (i) From Definition 2.5, we have

(ii) Since Ut,T - p ( t , 6-tw)B(6-tw) is closed, S ~ B ( W ) is closed as well. The following definition has been introduced in our paper [ll]in order to study large time behaviour of stochastic Navier-Stokes equations in unbounded domains but which has also other applications, see the above mentioned paper.

Definition 2.6. A RDS cp on a separable Banach space X is said to asymptotically compact (AC) iff for any bounded sequence { x ~ c} X ~ and any sequence { t n } c [ O , o o ) such that tn -+ ~ 0 the , set { p ( t n ,d-t,W)Zn : E N} is relatively compact in X .

be

The following result lists all the basic properties of the Q-limit sets, see [I11 section 3 for proofs.

Theorem 2.4. Suppose that cp is an asymptotically compact RDS on a separable Banach space X . Then for any bounded deterministic set B c X , IP-a.s., for all w E R, Q,(w) is nonempty, compact, strictly 9-forward invariant and attracts B .

83

Remark 2.5. In the deterministic setting the notion of asymptotic compactness was first introduced and studied by Ladyzenskaya in [33]. The first one to use it for studying asymptotic behaviour of 2-D NSEs in unbounded domains was Rosa in [35]. Rosa proved that 2D NSEs generate an asymptotically compact dynamical system on the space H , see below. Later on Ju in [31] showed that Rosa's results are also valid for the dynamical system generated by 2-D NSEs on the space V.

3. RDS generated by SNSEs We begin this section with listing the properties of the set D c R2 which we will require later on. We assume that D is an open (bounded or unbounded) set with sufficiently smooth boundary d D such that the following Poincar6 inequality (3.1) holds on D. More precisely, we assume that Assumption A . l . There exists a constant XI > 0 such that

In the inequality (3.1), V denotes the gradient operator, V u =

8'U - D (G? G)- ( 1%D2U)-

Also we need assumption for the existence of the following operator extension: Assumption A.2. There exists a linear and bounded operator ED : IH12(D) + IH12(R2)such that ED is a retraction, i.e., E D ( U ) ~=Du as., and for some positive constants, for simplicity we just denote by C, for u E W2(D),

5 cIuIL2(D),

IED(u)l!L2(WZ)

IvED('LL)ILz(Rz)

5 CIvulL2(D),

1v2E D ( u )1 LZ(R2) 5 C Iv2uIL2 ( D ), where lV2u1& = C;,,,,

IDjkui12LZ and

1Vu& =

c;,jIDjuiILz. 2

We consider the flow of an incompressible viscous fluid of constant density in D governed by the Navier-Stokes Equations: $-vAuf(u.V)u+Vp=f

u(.,O) = u g

in D , in D , on d D , in D.

(3.2)

84

Here v > 0 is the kinematic viscosity of the fluid and f : [0,m) x D -+ R2 is the external body force, u(t,x) E R2 and p ( t , x) E B denote respectively the velocity and the pressure of the fluid at time t 2 0 at point x E D. Let IL2(D):= L 2 ( D , R 2 and ) W1 := H1(D,R2)with inner products (., .) 1

and (., -)m~ and norms 1 . I := (., -)'. and 1 . \HI:= (., .)& respectively, where (21, U) := JD U .

(u,W)WI := C:=,JD V U .~Vvj dx

u,2, E IL2(D);

dx,

+ J,

. t~dx,

u,w E W'(D).

Set

v := {v E C?(D,R~): v . w = o in D ) , H V

:=

the closure of V in IL2(D),

:= the closure of

V in W'(D).

Remark 3.1. Denoting by JHIi(D) the closure of Cr(D,R2)in can show that V equals t o the closure of V in WA(D).

W1(D),one

Remark 3.2. As a consequence of Assumption A.l the norm on WA(D) (resp. on V )inherited from W1(D)is equivalent to the norm generated by the following scalar product

((u, v)) :=

c:=, J, v u j .

1 1 ~ 1 1 2 := ( ( % U ) ) , Moreover,

( 1 . ~ 1 1 2~ X1lwl2,

v), u,w E WA(D)(resp. v).

V V ~ ~ Z ,u,w E

Hh(D)(resp.

for all w E V .

By the Riesz Representation Theorem we can identify H with its dual H', so we have V c H = H' c V' with imbeddings being dense and continuous. We define a linear and bounded operator A : V -+ V' by (Au,w):= ( ( u , ~ )u,w ) , E V , where (.,.) is the duality product between V and V'. One can show that the map A is a linear isomorphism from V onto V' (for details see [40], p.249, p.24 and p.27). It is well known, see [18] and [40], that for D bounded and with C2class boundary, 1 Aul is a norm on VnW2(D),equivalent to the norm induced by IH12(D). In fact this remains true if D is a domain satisfying the Poincar6 inequality, see assumption A.l. We have the following similar result (see [12] Lemma 1). Lemma 3.3. Suppose that Assumption A.l holds, then IA. I is a norm in V n W 2 ( D ) . This norm is equivalent to the norm inherited from W 2 ( D ) . Moreover, D ( A ) c V c H = H' c V' and V = D ( A i ) .

85

We now recall a definition of a trilinear form b ( u , v , w ) = u(z)Vv(z)w(z) dz, whenever u,w,w E Lioc(D)are such that the integral on the RHS exists.

,s

Remark 3.4. Let V be the closure of V in Hi n L"(D). Then in fact b is a trilinear continuous form on v x v x V for any open set D c R". If D is bounded and n 5 4, b is trilinear continuous form in V x V x V . In particular, when n = 2, from Remark 3.1, it is easy to see b is trilinear continuous in V x V x V for any open set D c R2. Then from Lemma II. 1.2, Lemma III.3.3, III.3.4 and Lemma 1.1.3 in [40], we have the following properties of the form b (valid for any open set D c

R2), b(u,w,w) = 0, u E

v,

21

E

WA(D),

(3.3)

b(u,v,w) = -b(u,w, w), 21 E v , v , w E @ ( D ) , Ib(u,v,w)l 52?1uI?lluIIt11w111WIbIIWllt, u,w,w E HA(D), Ib(u,v,w)l

5

I~IL4(D)IV~IL~(D)IWIL4(D), U,V,W E

(3.4) (3.5)

HAP). (3.6)

If Assumption A.2 be satisfied, then there exists a positive constants C,, Cz such that

I~(U,V,W)~

5 C I I U I ~ I A Z L I ~ I IuVE~ ~ D~( A W)~, w , E V,WE H ,

Ib(u,v,w)l 5 C ~ I Z L I : I I Z L J I ~ I A Z ~u~E? V,w ~ ~ ~E~D~(~A~) W , wIE, H .

(3.7) (3.8)

If u,w are such that for some positive constant C , Ib(u,w ,w)l 5 CIIwII, w E V , then there exists a unique element in V', which we denote by B(u,w),such that ( B ( u , w ) , w )= b(u,w,w),w E V . Moreover, if for some C > 0, Ib(u,w, w)l 5 Clwl, w E V , then B(u,w) E H . The following inequalities follow from the inequalities (3.7) and (3.8) respectively:

IB(u,w)l 5 Cllul?IAulf11~11, u E D ( A ) , v E V, (B(u,v)lI:CZIU~~IIUII~II.UII~IA.U~~, u E V,VE D ( A ) .

(3.9) (3.10)

Furthermore, the inequalities (3.7) and (3.8) imply the following,

B(u,'u)E L 2 ( 0 , T ; H ) , u,w E L 2 ( 0 , T ; D ( A )nL"(O,T;V), ) (3.11) B ( ~ , E L ~ OT ;,v'), u, E L ~ ( o , TI L ; ~(D)). (3.12)

Remark 3.5. If D C R3,the equalities (3.3), (3.4) and (3.6) hold. Inequality (3.6) takes the following form: lb(u,w,w)l 5 21~1~/~11u11~/~(Iwlllw~

86

This is because the Sobolev inequality takes different form for n n = 3, i.e.

=

2 and

One can define a solution t o problem (3.2) as follows. Suppose f E Lfoc([O,oo);V‘) and uo E H . A function u E Lfoc([O,oo);V ) is called a solution to the NSEs (3.2) iff

or U’

+ VAU- B(u)= f,

(3.14)

u(0)= uo.

Similarly, if f E LfoC([O,co); H ) and uo E V , then a function u E LfOc([0,oo);D(A)) is called a solution to NSEs (3.2) iff (3.13) or (3.14) holds.

Remark 3.6. Note that if u E Lfo,([O,00); V ) (rep. u E Lfoc([O,m);D ( A ) ) ) then B(u) E L;,,([O, 00); V’) (rep. B ( u ) E L:,,([O, 00); H ) ) . Hence (3.14) makes sense, see Temam [40] section 111.3 for discussion. From now on we will be interested in the NSEs in the functional form (3.14) with the external force f perturbed by an additive noise, i.e.

+

du [VAU- B ( u ) ]d t = f d t 4 0 ) = uo,

+ dW(t),

t 2 0,

(3.15)

where W ( t ) t, E R is a two-sided cylindrical Wiener process in some Hilbert space K (which will be specified below). Let us recall, see [14], that given a probability space U = ( a , F , P ) , a K-cylindrical (i.e. with identity I as the covariance operator) Wiener process, is a pair ( ( 3 t ) t E R , (W(t))t?o) consisting of a filtration (Ft)tEl and a family (W(t))t?o) of bounded linear operators from K into L2(R,3,P) such that: (i) for a11 t 2 0, and +, cp E K, [W(t)$JW(t)cpl= t($,V)K, (ii) for each $ E K, W(t)+,t 2 0 is a real valued (Ft)-adapted Wiener process. We will need one or both the following two assumptions

87

Assumption A.3. K is a Hilbert space such that K 6 E (0,1/2), the map A-6 : K

-+

C

H nIL4 and, for some

H n IL4 is y-radonifying.

(3.16)

Assumption A.4. K is a Hilbert space such that K C D ( A ) and, for some 6 E (0,1/2), the map A-6-1 : K -+ H is y-radonifying.

(3.17)

Remark 3.7. Let us denote by X, the closure of V in V ( D ) . If D is bounded, then IL4 L) IL2. Hence, if the map A-6 : K + X, is y-radonifying, then the Assumption A.3 is satisfied. Theorem 3.8. Assume that the assumption A.3 is satisfied. Then there exists a metric DS Z = (R,F,P,t9), a K-cylindrical Wiener process ((Ft)tEw,(W(t))t2,) and a family z,, a E N of measurable maps from R x R to H n IL4 such that the following conditions are satisfied.

(a) For each a E N,z a ( t ) , t E R is a Gaussian, stationary and continuous HnIL4-valued process and for each w E R and a E W one can find C,(w) > 0 such that

Iz,(~)/H+L~

I C a ( w ) ( 1 + It11'2), t E

(3.18)

(b) The process z,(t), t E JR is an (Ft)-adupted mild solution of the problem dz

+ ( v A+ a ) z d t = d W ( t ) ,

t

E

R.

(3.19)

(c) The process z,(t), t E R} has a unique, hence ergodic, invariant measure P, and

(d) For every a E N and all w exist

E

0, the following limit (with X E {H,IL4})

(3.20)

(e) If in addition Assumption A.4 is satisfied, then the metric DS Z can be chosen in such a way that the trajectories of the processes z, belong to LTo,(JR;D ( A ) )n L"(R; V ) and (d) holds also with X = V .

88

In order t o define a solution t o the problem (3.15) we notice that on the informal level, if u(t),t 2 0 is such a solution and z,(t), t 2 0 is the process from the above Theorem, then w(t) := u ( t )-z,(t), t 2 0 , is a solution (with 210 = 2 - Z " ( 0 ) ) to

f =-vAw+B(w+z)+az+f, v(0) = wo.

(3.21)

Remark 3.9. The above problem (3.21) can be treated for any fixed deterministic function z : [o, co)+ I L ~ ( Dn)H . Hence, in order to define correctly a solution to problem (3.15) we need firstly to define a solution to (3.21) and secondly we need to show that the resulting process u = v z , does not depend on the choice of a. We begin with the former issue.

+

Definition3.1. Suppose z E L4([0,T];IL4(D)), f E L2(0,T;V')andvo E H . A solution of problem (3.21) is a function w E L2(0,T;V)such that the following equality holds in V' sense: t

w(t) = wo--v1 A v ( s )ds +

1 t

+

B ( v ( s ) z ( s ) )ds (3.22)

Remark 3.10. Note that if z and w satisfy the conditions from Definition 3.1, the continuity properties of B imply that B(v+z) E L2(0,T ;V ' ) . Since also for w E L 2 ( 0 , T ;V ) ,Aw E L2(0,T ;V ' ) ,we infer that w' E L 2 ( 0 ,T ;V ' ) . This then implies, see Lions and Magenes [34] I, p.238, that w E C([O,T]; H). The second issue is discussed in detail in section 5 of our paper [la]. For a given a 2 0 and uo E H let v" be the unique solution (see Theorem 3.14 below) of the problem (3.21) with initial condition v"(0) = uo - ~ " ( 0 ) We . have

Proposition 3.11. Suppose a and ,I3 are two nonnegative numbers and 210 E H . Then, for all t E [O,T]and all w E R , v"(t) + z a ( t ) = d ( t )+ z p ( t ) . Finally we are ready to define a solution to problem (3.2). Definition 3.2. A solution to problem (3.2) is an (&) adapted L4(D)n Hvalued process u ( t ) ,t 2 0 with H-valued continuous paths such that for some a 2 0, u ( t )= v"(t) + z a ( t ) ,where w" is a solution of problem (3.21) with wo = uo - ~ " ( 0 ) . Theorem 3.12. ( [34], Theorem 3.4.4, I, p.238, Theorem 4.3.2,lU, P.22 1

89

(i) If f E L2(0,T ;V’) and uo E H then there exists a unique u E L2(0,T ;D ( A ) ) such that u’E L2(0,T ;H ) (so that u E C([O,TI;H ) ) and u solves the problem

+

u’ vAu = f, u(0) = UO.

(3.23)

(ii) If in addition f E L2(0,T ;H ) and uo E V , then the unique u in part (i) is such that u E L2(0,T ;D ( A ) ) and u’E L2(0,T ;H ) (so that u E C([O,TI;V ) ) . Theorem 3.12 (ii) implies the existence of a constant CT

> 0 such that

for all f E L2(0,T; V’) and uo E H . But, we have a sharper result.

CoroZZay 3.13. If u solves problem (3.23) with f E L2(0,T ;H ) and uo E V, then

+

v21~12Lz(o,T;o(A))

5 3vIIuoI12+ 5 1 f i E ~ ( ~ , ~ ; ~ )(3.24) .

The proof of Corollary 3.13 relies on the following a’priori estimates in V , see [12] for the details,

d

If Wl2, t 2 0 , -llu(t)l12 + vlAu(t)12 = 2 ( f ( t ) , 4 t > )I 7 dt

(3.25)

which is a consequence of a V-norm version of Lemma lU.1.2 in [40]. The following theorem can also be proved by the classical Galerkin approximation method (see [ll]), but motivated by [8], we have a different proof which is based on the Banach fixed point theorem.

Theorem 3.14. Let us suppose that D A.2.

c

IK2 satisfies Assumptions A.1,

(i) Ifz E L 4 ( [ 0 , T ] ; H n I L 4 ( D ) )f , E L2(0,T;V’) and wo E H , then problem (8.21) has an unique solution in the sense of Definition 3.1. Moreover, the map ( L 4 ( [ 0 , T ] ; H n I L 4 ( D ) ) ) x L 2 ( 0 , T ;xVH’ ) 3 ( z , f , w o ) Hv

E L2(0,T;V)nC([0,T];H)

is continuous. I n particular, f o r fixed z and f and t E IO,T],the map H 3 vo H v ( t ) E H is continuous.

90

(ii) If in addition, z E C(R;V) n L~o,(R;D ( A ) ) ,f E L2(0,T ;H ) and wo E V , then the unique solution v from part (i) is such that w E L2(0,T; D(A)) and w' E L2(0,T ;H ) (and hence v E C([O,TI;V ) ) . Moreover, the map (C(a;V)nL~~,(W;D(A))) x L 2 ( 0 , T ; H )x V 3 (z,f,vo) H v E L 2 ( 0 ,T; D(A)) fl C([O,TI;V ) is continuous (in fact, real analytic). In particular, for fixed z, f and t E [O,T], the map V 3 wo H w ( t ) E V is continuous. Proof. Here we only give the proof of part (ii). The proof of part (i) follows then the same method as the proof of Theorem 3.1, Step 3 in [8]. To be more precise, we approximate wo E H by a sequence {vo,,} of elements of V , and we approximate f by a sequence { f n } of elements of L2(0,T ;H ) and approximate z by a sequence {zn} of elements from L"(0, T; V) nL2(0,T;D ( A ) ) . In view of part (i) there exists a unique solution w, of problem (3.21) with the regularity as therein. We then show that {wn} is a Cauchy sequence in L2(0,T ;V )n C([O,TI;H ) and the limit v of that sequence belongs also to L2(0,T ;V') and solves equation (3.21). The smooth dependence of solutions on the data, i.e. on f ,vo and z , in part (ii) follows from principle that a uniformly contractive family of mappings dependents smoothly on the parameter, see e.g. [30] section 1.2.6, see also [4] and [lo]. The continuous dependence on the data in the case (i) is proved in Lemma 5.3 in section 5. In fact only continuity in vo is partially proved and formulated there but the proof can be easily adjusted to cover also continuous dependence on uO,t and z . Local existence. Since z E C(R;V), T > 0 such that, R := S U P ~ S Ilz(t))I ~ ~ T < 00. Let X T = { U E L ~ ( o , T ; D ( A ) ) , E u 'L2(0,T;H ) } . Then with the norm IuI$, = ~ 2 / ~ l ~ ~ ( o , T ; D ( A ) ) + IXuT' is ~~ a Hilbert ~(o,T;H), space. If u solves the problem u' + VAU= B(v z ) + LYZ f with initial data u(0) = V O , define a map @T : X T --+ X T by @ T ( w ) = U. As a direct consequence of (3.11), we have the following Lemma:

+

+

Lemma 3.15. If 0 < T < 00, then for any w E X T , f E L2(0,T;H ) and z E L2(0,T ;D ( A ) )n L"(0, T ;V ) ,B(w z ) f E L 2 ( 0 , T ;H ) .

+ +

From Theorem 3.12 and Lemma 3.15 we infer that @,(v) E X T for v E X T . Hence, ~ D T is well defined. Let us denote XT,R,~,,= {W E X T : I U I X , < R, v ( 0 ) = vo}. We have the following two crucial Lemmas:

91

Lemma 3.16. If u E XT,then

Sketch of the proof of Lemma 3.16. We use the a'priori inequality (3.25) in V-norm and the following a'priori inequality for the H-norm: 0 _ _:t142 ~11412 = (f7u) 5 ;11~112 +

:

klfl"

+

Let us now fixT1 such that 0

< TI < co and put R1 := supo &IIvoll and T 5 7'1, then there exists a constant C ( T )= C(T,R, R1) such that I@T(v1)- @T(w)IxT I C(T)I'Q- ~ z I x T f o r ~. for fixed R and R1 C ( T ,R, R1) + 0 as T \ 0. q , v 2 E X T , R , ~Moreover,

Sketch of the proof. Let 211, vz E X T , R , ~Denote ~. ui = @T('u1), i = 1,2. Then u = u1 - uz E XT satisfies: u' + uAu = B(v1 z ) - B('u2+ z ) and u(0)= 0. Hence by Corollary 3.13 we infer that

+

92

Put

=

(

2.701/221/4R1 31/ZY3/4

+ 7 ) T 1 j 4 . Then IulxT 5 C(T)Ivl 51/223/4R

0

VZIXT.

Therefore, in view of Lemmata and 3.17 and 3.18 there exists To > 0 9 2 (e'g. One can take To = min{T2, ( ~ . ~ 0 1 / 2 2 1 / 4 R 1 + 1 ~ 1 / 2 2 3 / 4 y 1 / 2 ) 4 such that C(T0, R, R1) < 1 and for any T 5 TO,the map @T : XT,R,~,, 4 X T , R , is~ ~ a strict contraction. Therefore, by the Banach Fixed Point Theorem, there exists a unique w E XT such that @ T ( v ) = w. Obviously, this w is a solution of problem (3.21) on [0, TI. Global existence. The proof of the global existence is based on two a'priori estimates, one in the H-norm, one in the V-norm. We formulate these as two parts of the following lemma.

Lemma 3.19. Suppose that the domain D

c R2 satisfies Assumption

A.1.

(i) Ifz E L2(0,T;V')nL4(0,T;L4(D)),f E L2(0,T;V') and vo E H , and if v E L2(0,T; V) is a solution to problem (3.21), then

I

I V ( T > ~ ~Iv(0)12eJl-

+

A ~ ) ~ ~h(r1e-f; A

~ dr,)

o~_
0,

1

-a

lim

a V

T

lG,(s

+a)

-

IV[~I(~,.)

5

121111

5 CIIvII, we have

f i , ( ~ ) l & ( ~ , . )ds 4 0,

as a

\ 0.

(5.6)

Furthermore, from Lemma 3.3 and (3.35) we get that is bounded in L"O(T,O;W1(D,))nL2(T,0;W2(D,)).

{2),l~,}

(5.7)

Therefore, by Theorem 5.2, with X = W",D,), Y = WA(D,), we infer that the sequence {O,~D,}, is relatively compact in L2(T,O;WA(D,)). From Lemma 3.20, using a diagonal argument, we can find a subsequence c {G,} and some VO E L2(T,0; D ( A ) )n L " ( T , 0; V ) such that {&I}

t2,)

+ 60

V,!

-+ 60

21,,1~,.

-+

GOID,

weak-star in Lw ( T,0; V ) weakly in L2(T,0; D ( A ) ) strongly in L2(T,0;Wi(D,)), for each

T

> 0.

(5.8)

Therefore, we can pass to the limit in the equation for V,,, and hence conclude that 60 is a solution of the same equation with initial data x at time T . By the uniqueness of the solution we infer that 60 = 6. And by invoking a standard contradiction argument we deduce that the whole sequence (6,) converges to 6 in the sense of (5.8). This proves property (5.4). By (5.8), 2lnl(t)(D , + 6 ( t ) ( D , strongly in L2(T,O;Hi(D,)),for all T E N, and a.e. t E [T,01. Hence for all $ E V ,

( ( h ( t >Q),)

+

((fi(t),$)),

t E [T,01

Since { ((6,(t),Q))} is uniformly bounded and uniformly continuous on [T,01, we infer that ((fin/(t),$))

+

((G(tL$)), t E [T,01.

And since V is dense in V , we have 6,) ( t )-+ 6 ( t ) weakly in V . Finally, by the smooth dependence of solutions on the data, see e.g., [30] section 2.3.5, also see [4] and [lo], using a contradiction argument we deduce that 0 for the whole sequence { f i n ) , 'Un(t)+ 6 ( t ) weakly in V . The following Lemma follows from part (i) of Theorem 3.14, here we give an independent proof.

Lemma 5.3. Assume that D satisfies Assumptions A.1 and A.2. Suppose also that T < 0 and z E L2(T,0; V')nL4(T, 0; IL4(D)).Ifx, + IC strongly in H , thenv(.,T;z,) + w(.,T;z)i n C ( [ T , 0 ] ; H ) n L 2 ( T , 0 ; VIn) .particular, v(0, T ;z), -+ v(0, T ;x) in H .

101

Proof Let us use the same notations as in the proof of Lemma 5.1, then the function y n ( t ) = G n ( t ) - G ( t ) solves problem

c

d

+

+

VAY, = B(&) - B(G) B ( y n ,). Yn(0) = 2, - 2. ZYn

+ B ( z ,Yn),

By employing a similar argument to the one used in the derivation of (3.29), there exists C > 0 such that for all n E W and t E [T,O], rt

On the other hand, by part (i) of Lemma 3.20, the sequence (6,) is bounded in L 2 ( T , 0 ; V ) .Then by the assumptions z g L4(T,0;L4(D)), SUP, J; (llG,(~)11~ 11~(s)11~)ds I C . By (5.9),we have

+

t

1yn(t)I2I 12, - $I2

+

+ h(~)1 I Y1 ~ (~ S )) ? ~t ~E, [7‘,01,

(/~(s)&(D) T

Using the Gronwall Lemma we infer that Iy,(t)I2 I eCTIx,-x12, n E N, [T,O].This implies that S U ~ ~ ~lyn(t)I2 [ ~ , -+ ~ ]0 as n 00. This fact, together with the inequality (5.9),easily implies that vs,” lly,(~)11~ ds 4 0 as n 4 00,what concludes the proof of the Lemma.

t

E

-+

Lemma 5.4. Assume that D c R2 satisfies Assumptions A.1 and A.2 and z E L2(T,0; D(A)) n LO”(T,O;V).Suppose that z, 4 x weakly in V and strongly in H and let T < 0 be fixed. Let us denote Gn(s)= v ( s ,T ;x,) and G(s)= v ( s , T ; o )for s E [T,O].Then limsup ,-too

LO

e”X’“b(G,(s),G,(s),AG,(s))

ds

(5.10)

=le~X1’b(G(r),i.(s),AG(s))ds,

LO l

limsup

eUX1”b(i.,(s), z ( s ) ,AG,(s)) ds

,-too

=

eUX1”b(G(s), z ( s ) , AG(s))ds,

(5.11)

102

lim sup n+cc

=

l

euXls b( z ( s ) , Gn(s),AG,(s)) ds

(5.12)

eVx1"b(z(s),G(s), AG(s))ds.

Sketch of the proof. We can prove this Lemma by applying inequalities 0 (3.5), (3.7) and (3.5) and Lemma 3.20, Lemma 5.1, Lemma 5.3. 6. Asymptotic compactness of the RDS generated by the NSEs Part (i) of the following theorem is proved in [ll],while a complete proof of part (ii) will be given in [12]. Here we are only concerned with part (ii).

Theorem 6.1. Assume that the domain D satisfies Assumptions A.1 and A.2.

(i) If in addition Assumption A.9 is satisfied and f E V', then the RDS generated b y (3.15) on H is H-asymptotically compact. (ii) If Assumption A.4 is in addition satisfied and f E H , then the RDS generated by (3.15) on V is V-asymptotically compact. Proof of part (ii). The argument presented below is done for a fixed "good" c V be a bounded set. In view of part (ii) of T h e e rem 4.1 there exists a closed and bounded set K ( w ) c V which absorbs B. Therefore there exists N B ( w ) E W, such that for any n > NB(w), 'p(t,,6-t,w)B c K ( w ) . By the asymptotic compactness of 'p in H , see Proposition 3 in [ll],for a given sequence {x,} c B , there exists a subsequence of {x,}, denoted again by {xn} and YO E H such that

w E 0. Let B

'p(t,,

29-t,u)zn

YO,

-+

strongly in H.

(6.1)

Since K ( w ) is bounded and closed in V , it is also weakly compact in V . Therefore, analogously to the proof of Proposition 3 in [ll],we can find a scale of subsequences . . . c {n(k)}r=o c {n(k-l)}r=o c . . . c {n(o)},"=oc W and a V-valued sequence {yk}F!?o, such that for k 2 0,

'p(-k

+ tn(k),d-tn(k)w)x,(k)

+ yk

weakly in

v,

as n(k)+ 00.

(6.2)

as n(O)+ a.

(6.3)

In particular, 'p(t,(o),

29-tn(o)

w)z,(o)

-+

yo

weakly in V,

By an easy abstract argument we can show that (6.1) and (6.3) imply that $0 = yo. Hence, in particular, YO E V .

103

Moreover, denoting the weak limit in V by wv-lim, by the weak continuity of the RDS as given in part (ii) of Lemma 5.1, since {n'')} c {n(O)}, we have

since V is a Hilbert space, (6.6) in conjunction with (6.7) will imply that p(tfi,fl-tau)xfi - z(O) 4

-

z(O)

strongly in V.

Therefore cp(tfi,fl-t,w)zfi 4 yo strongly in V, what would imply that the set {p(t,, fl-t,u)x,) is relatively compact in V . Hence, in order to show that the RDS generated by Equation (3.15) is asymptotically compact in V, we only need to show that (6.7) holds true. Following the idea from [35] we define the bilinear form [I., .I] : D ( A ) x D ( A ) + R by

[~u,wI]

= v(Au,AzI)-

1 -2~ X ~ ( ( U , W ) ) ,

U,W E

D(A).

One can easily see that [I., .I] is a scalar product on D ( A ) and that the norm generated by it, i.e. [1.,.1] = [Iu,uI] = vlAz~1~ - ivXlllul12 is equivalent to the norm IA(.)I in V n H 2 ( D ) .Indeed, we have [/.,.I] 2 vlAu12- i v l A ~ = 1~ ?jvlAu12.Thus ?jvlAu125 [lull25 vlAu12. If w is a solution on some interval [to,co) to problem (3.21) with z ( . ) = z ( . , w ) . Then by (3.30), dt

104

which is a linear differential equation in the function t H llv(t)1I2.Therefore, by uniqueness of solution t o such an equation, we have for t T t o ,

> >

(6.8) Then by the cocycle property of RDS generated by problem (3.15) and denoting v(s, -k; w , p(tn(k)- k , %tn(k)w)xn(k) - z ( - k ) ) by w n ( k ) ( s ) ,we infer that

Let Gk = yk - z ( - k ) and ' u k ( S ) = ? J ( S , - k ; w , & ) , s 2 -k. Since eVX1'(cxz(.)+f), e U X 1 ' B ( z ( .E) )L2(-k,0; H ) , from (6.2) and (5.4) in Lemma 5.1, we have

=

J_", e v X l S ( a z ( s4)-f,A v k ( s ) )ds. limsup

e v X 1 s ( B ( z ( sz)(,s ) ) Avn(k) , ( s ) )ds

(6.11)

n(")+mf k 0

e V X l s b ( z ( sz) (, s ) ,A v k ( s ) )ds. = l k

In view of (6.2), (5.3) in Lemma 5.1 and applying Fatou lemma, we infer

105

that

Hence

From (6.2), for a fixed k , we have (P(tn(k)

- k,d-t,(k)W)xn(k)- z ( - k )

+y

k - z(-k)

weakly in V, as n

+ 00.

By the cocycle property of cp we infer from (6.1) that (P(tn(k) -

k , d - t n ( k ) W ) ~ n ( k )- x ( - k )

+yk -

x(-k)

strongly in H .

Applying next Lemma 5.4 to (6.9), (6.10), (6.11), (6.12) we get, for fixed

k, limsup IIv(tn(k),d-t,(k)W)2n(k)- z(O)1l2 5 n ( k ) -00

+ p:,

where

+

- 0 : Xt @

A}. The killed symmetric stable process X D in D is defined by

x,"= { X8,t ,

i f t TD

where d is the cemetery point. The Green function G D ( x , ~of) X D is a function that is continuous on D x D except along the diagonal such that for every Borel measurable function

f 2 0 on D,

When D is C1-smooth, at the first exit time from D, the symmetric astable process X starting from x E D lands in the exterior of D almost surely and the exit distribution is absolutely continuous with respect to the Lebesgue measure in on D x

D".That is, there is a measurable function K D ( ~z ),

Dc such that for every Borel measurable function 'p 2 0 on Dc, EZ

['p(xTD)l

=

DC

KD(x,Y)P(z)dz.

The function K D ( z y) , is called the Poisson kernel of X on D. Recall that an open set D in R" is said to be C1il if there is a localization radius R

> 0 and a constant A > 0 such that for every Q

a C171-function $ = $Q : R"-l

-+

R satisfying $ ( O ) =

8D,there is 0, ~ ~ V $5~A,~ m E

- V$(z)I

5 Alx - zI, and an orthonormal coordinate system y = ( ~ 1 ,. .. ,~ " - 1 , ~ n = ) (Y, Y") such that B(Q,R) n D = B ( Q ,R ) n {Y : yn > $(c)}. It is well known that for a bounded C1)'domain D , there exists ro > 0 depending only on D such that for any z E d D , 0 < r 5 ro, there exist two balls B f ( r ) and B,"(r)of radius r such that B f ( r ) c D, B,"(r)c R" \D and { z } = d B f ( r ) n aB,"(r). lV$(x)

The following sharp estimates on the Green function GD were obtained independently in Chen and Song [3] and in Kulczycki [ 5 ] .

127

Suppose that D is a bounded C1il domain in R". Let b(x) = d(x,a D ) be the Euclidean distance between x and aD. Then there exist constants C1 = C l ( D , a )> 0 and C2 = C z ( D , a ) > 0 such that for x,y E D ,

In Chen

[a], the following sharper estimate is given.

Suppose that D is a bounded C1vl domain in R". Then there are constants c1 = q ( D ) > 0 and c2 = q ( D ) > 0 , depending on D only, such that forx1y E D ,

We remark here that the exact lower bound estimates in (1)-(2) for Green function GD are more difficult t o establish than the upper bound estimate. The novelty of the estimate (2) over (1) is the more precise and explicit information on how the constants C1 and C2 in (1) depend on a , which can be used to recover the Green function estimate for killed Brownian motion in D

c R" when n > 3.

Note that since the symmetric a-stable

process converges weakly in D( [0, m),R") t o a Brownian with infinitesimal generator A as a

T 2 (cf., e.g.

Ethier and Kurtz [4]), the Green function

G D converges pointwise to the Green function of the killed Brownian mo-

tion in D. Here D([O,m),R") is the space of right continuous R"-valued functions with left limits, equipped with the Skorohod topology. Thus, as is pointed out in Chen [2], by letting a

T 2 in estimate (2) we immediately

get the following well known sharp estimates on the Green function of the killed Brownian motion, whose upper and lower bound estimates were first obtained by Widman [6] and Zhao [7],respectively.

128

Suppose that D i s a bounded C1il domain in R" with n 2 3 and that GD is the Green function of the killed Brownian motion in D. T h e n there exist constants c1 = q ( D ) > 0 and

c2 = cz(D)

> 0 such that for all x , y

E

D,

In Chen [2], the estimate (2) is first proved for balls, which is then used to establish the estimates on general bounded Clil domains. The proof for the lower bound estimate of G D in (2) for a bounded C1il domain D is fully given in Chen

[a], while for the upper

bound estimate it was claimed

that it can be obtained by following the proofs of (1.4) and (1.5) in Chen and Song [3],keeping better track of all the constants and using the refined estimates of the Green functions and Poisson kernels for balls obtained in Lemmas 2.2 and 2.3 of Chen [2], However, the original proof of (1.5) given in Chen and Song [3] contains a gap in that it works only for rather than for all

Q

cy

E

(0,l)

E (0,2) as claimed there. (We thank Piotr Graczyk

for pointing out this gap.) The purpose of this note is to present a proof for the upper bound estimate in (2).

Theorem 0.2. Suppose that D i s a bounded C1il domain in R". T h e n there exist a constant c = c(D) depending only o n D such that for all X,Y

E D,

Lemma 0.1. There exists a constant c = c ( n ) > 0 such that for any a E R" and r > 0,

where G B ~ ( i~s the , ~ )Green function of B c ( a , r ) = {x E R" : Ix - a1 > r } and b ~ = ( ~ , ~i s) the ( x )Euclidean distance between x and a B ( a , r ) .

129

Proof. By Lemma 2.2 of Chen

[a],there is a constant

c = c(n) 2 1 such

that

The rest of the proof is exactly the same as that of Lemma 2.5 in Chen and

) ( Z0, Song [3], provided we use the above improved estimate on B B ( ~ , ~y).

The next result is a strengthened version of (1.4) of Chen and Song [3].

Lemma 0.2. Suppose that D i s a bounded

exists a constant c = c(D) depending only

Proof. Let xo E dD be such that Ix

domain in R". T h e n there

D such that for all x,y

on

- 201

E

D,

= 6 ( x ) . Consider the ball

B = B,"' ( T O ) = B ( a , T o ) . It follows from Lemma 0.1 that there is a constant c1 = q ( n )> 0 such that

= ccr

Iy - u p / 2 0

p

6(x)"/2 12 - y1n-P

5 ccr

diam(D)

+ ro

TO

6(~)"/~ I Z - yl"-"/2

where diam(D) stands for the diameter of D.

' 0

The following is a refinement of (1.5) in Chen and Song [3].

Lemma 0.3. Suppose that D is a bounded C1il domain in R". T h e n there

exists a constant c = c(D) depending only o n D such that f o r all x , y

Proof. If b(y)

2 ro or S(y) < ro and 1x - yI I 8S(y), then

E

D,

(4) follows from

(3). By the symmetry of the Green function Go, (4) holds when b(x) 2 ro

< T O and Ix - y ( 5 8S(x). So we assume that 6 ( ~ 8max{d(x), b(y)}. Set r = min{lx - y1/8, T O } . Let ZO,

or b(x)

130

yo E

all be such that Iz - zol = S(z)and Iy - y o ]

= S(y),

respectively. Let

B(a,r ) = B,"'(r) and B(b,r)= B p ( r ) . Without loss of generality, we can assume that b is at the origin and that yo = (-r,O,... ,O). In the sequel, c is a positive constant that depends at most on domain

D but its value

may change from line to line. We remark here that dimension n is a part of information about domain D so c may depend on dimension n as well.

( z0 ,on B ( a , r ) , Clearly, as G B ~ ( ~ , ~.))=

Note that, since

IC

-

yI

2 8r, for u E B(yo,4r),

while by Theorem A of Blumenthal, Getoor and Ray [l]and the scaling property for symmetric a-stable process X , KB(yo,4r)(% u)

Here

r is the Gamma function defined by r ( X )

=

tx-le-tdt

for X

> 0.

We note that there is a constant c > 1 such that n-a!

sin - < c a ( 2 - a ) for 0 < a! 5 2 , 2 -

(8)

and

1

-< xc -

r(x)5 Cx

for

o 1, with respect t o the norm

{ E ~ denotes }

the canonical basis in Rd.

139

If we consider maps Z p ( 7 ) E Tp(,)(M) such that Z ( T ) = tE+,(Z(T)) belongs t o the Cameron-Martin subspace H of the Wiener space, then we can also define derivation along the “vector field” 2 by

1

1

DzF

=

D,,,Fi@(T)dT

Differential calculus on the path space of a Riemannian manifold can be “transported” to differential calculus on the Wiener space through the It8 map. The price to pay is that Cameron-Martin tangent space is not preserved. This phenomena leads to a necessary extension of the tangent space and the definition of the so-called tangent processes (cf. [lo], [4]). The corresponding result is the following:

Theorem 2.1. (Driver [lo], Fang-Malliavin [I21 and Cruzeiro-Malliavin 141) A scalar valued functional F defined o n the path space Pmo(M)is differentiable along a n adapted vector field 2 i f and only i f F o I is differentiable o n the Wiener space along a semimartingale given by


’) := X(XGp) - 1) and the associated semigroup Tin’’) := etL(””);n = 1. . . ,+cc

Lemma 4.2. For gn E D((L(”))’), assume that SUP II(L(n))29nIILZ,< 03, n

then f o r fixed T > 0, Tin”)gn converges uniformly (with respect to n and E (O,T])to Tin)gn in L i as ,D + 00.

t

145

Proof. Notice that

On the other hand,

Thus

Therefore

(/

03

IIL(niX)gn - L(n)gnllL; 5

e-'s ds) II(L(n))2gnIIL;/X 5 C/X,

0

which yields the result.

0

Theorem 4.3. Let gn E L i be such that ing, T > 0, we have

+g

in L2. Then for fixed

Proof. We first prove this for special g and gn. For h E L 2 , set g =

( I - L)-2h = (G1)2h, gn

=

( I - L ( n ) ) - 2 ( j n h= ) (Gp))2(jnh).

Then from Proposition 4.1, we know that lim IJi,g, - g ) ) L z = 0.

n-oo

146

Hence

by Lemma 4.2 and Hille-Yosida approximation, changing the order of the limits we obtain

where we used Lebesgue dominated convergence theorem in the Set Then

First letting n --+

00,

and then k

4

00,

we get

lim IlinTt(n)gL- TtgllLz = 0.

n-m

Finaly, for any gn E L i , if

147

Then

0

and the proof is completed. 5 . The lifted semigroup

In [4] a Markovian connection on the path space was introduced in order to renormalize the Levi-Civita connection, which produces a divergent curvature. If 21, 2 2 are adapted vector fields, and zi(.) = t ~ + . ( Z ( . )i)= , 1,2, are the corresponding Cameron-Martin vectors, the Markovian connection is defined by

where R denotes the curvature tensor of the manifold M and odx stands for Stratonovich stochastic integration. Here we have identified the covariant derivative with its image through the parallel transport. In [7]we have defined on the finite manifold M," a Markovian connection which is Riemannian : for any smooth vector fields Y,2 E T ( M r ) ,we put

d -(v",)x(v, ds

s-)

where s- = mux{si 5 s}. The operator L" can be lifted to the frame bundle O ( M F ) through the connection V", thus defining an operator C&M,n, such that, for any smooth function f, 'c&M,-,

(f O ).

=

(L"f)

O

7r

where 7r denotes the canonical bundle projection. Furthermore, if 2 is a vector field on M," and Fz(r) = r - l ( Z ) E H" denotes its scalarizition, we have: C"OM,n)FZ = F P Z , where

148

On the other hand, an operator C on vector fields on the path space associated t o the Markovian connection was defined in [5] (cf. also [3]). For cylindrical vector fields Z E C(H) = { Z ( p ) = Eel Fi(p)hi,Fi cylindrical }, where { h i } denotes a basis in H, it can be written as:

a

I

1

J’

CZ =

1

V:,,Zdr

0

-

V,),Z

o

dxa(r)

01

In [7] we have proved the following:

Theorem 5.1. For any 2 E C ( H ) we have

( P+qz,+ (C+ I ) Z in L2(Pmo(M), q H ) , where

Dirichlet forms can be naturally associated in this framework to the operators C n I and C 1 and in the correspondent resolvents converge weakly in L2. Concerning the semigroups, we have constructed in [7] a process rt = ( p t , e t ) on the space Pmo(M)x P ( O ( d ) )as the lift of the O.U. process p t on the path space through the Markovian connection. The results that follow were proved under the assumption that the Ricci tensor of the manifold M is zero. Although this is very restrictive, under suitable modifications of the connection and of the metric on the path space (as in [3]) they can still be stated without this assumption. A representation formula for the semigroup associated t o C 1 holds:

+

+

+

Theorem 5.2. For any 2 E C ( H ) we have

(T,“+”z)(~)= eMtE(r-l(w,p, t)z) and, concerning the convergence of the semigroups, we have:

Theorem 5.3. Let Z E C ( H ) , 2, E L2(MF,vn,,;H2). T h e n for any Y E L2(Pm,(M), v ;H ) we have

E”((y+’) ZnlY)H) E”((T:+rzly>H> +

This weak convergence can be improved by the methods described in paragraph 4 and strong ( L 2 )convergence follows.

149

Acknowledgement: The first author wishes t o thank the organizers of the First Sino-German Conference on Stochastic Analysis held in Beijing for their invitation.

References [l]Andersson(L.) and Driver(B.), Finite Dimensional Approximation to

Wiener. Measure and Path Integral Formulas o n Manifolds. J . Funct. Anal, 165 (1999), 430-498. [2] Bismut (J. M.), Large Deviations and the Malliavin Calculus . Birkhauser, Basel , 1984. [3] Cruzeiro(A. B.) and Fang(S.), Weak Levi-Civita Connection f o r the Damped Metricon the Riemannian Path Space and Vanishing of Ricci Tensor in Adapted Differential Geometry. J . Funct. Anal., 185 (2001), 681-698. [4] Cruzeiro(A.B.) and Malliavin(P.), Renormalized Differential Geometry on Path Space, Structural Equation, Curvature. J.Funct. Anal., 139(1996), 119-181. [5] Cruzeiro(A.B.) and Malliavin(P.), Frame bundle of Riemannian path space and Ricci tensor in adapted differential geometry. J.Funct. Anal., 177(2000), 219-253. [6] Cruzeiro(A. B.) and Malliavin(P.) , Stochastic Calculus of Variations and Harnack inequality on Riemannian Path Space, C.R. Acad. Sciences Paris, Ser I, 335(2002), 817-820. [7] Cruzeiro(A.B.) and Zhang(X.), Finite Dimensional Approximation of Riemannian Path Space Geometry. J.Funct. Anal., 2003. [8] Cruzeiro(A.B.) and Zhang(X.), O n Harnack and Littlewood-PaleyStein inequalities for 1 < p 5 2. preprint, 2003. [9] Cruzeiro(A.B.) and Zhang(X.), A Littlewood-Paley type inequality on the path space. preprint, 2003. [lo] Driver(B.) , A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J.Funct. Anal., 110 (1992), 272-376. [ll]Driver(B.) and Rockner(M.), Construction of dzflusions o n path spaces and loop spaces of compact riemannian manifolds. C.R. Acad. Sciences Paris, Ser I , 320(1995), 1249-1254. [12] Fang(S.) and Malliavin(P.), Stochastic analysis o n the path space of a Riemannian manifold. J.Funct. Anal., 118 (1993), 249-274. [13] Li(X.D.), The de Rham-Hodge semigroup o n the path space: existence

150

and L2-contractivity. C.R.Acad. Sc. Paris, t.328,Sbie I, no 15, P.161166,1999. [14] Kallenberg (0.):Foundations of modern probability, Springer-Verlag, 1997. [15] Kazumi(T.), Le processus d 'Omstein- Uhlenbeck sur l'espace des chemins et le probleme des martingales. J.Funct. Anal. , 144 (1997), 20-45. [16] Kusuoka(S.), Dirichlet forms and diffusion processes on Banach spaces. J.Fac. Sci. Univ. Tokyo Sec I A 29 (1982), 79-95. [17] M a (Z.M.) and Rockner (M.), A n introduction to the theory of (nonsymmetric) Dirichlet forms. Berlin, Springer-Verlag, 1992. [18] Malliavin(P.), Formule de la moyenne, calcul de perturbations et theorkme d'annulation pour les formes harmoniques. J.Funct. Anal., (1974) 274-291. [19] Malliavin(P.), Stochastic Analysis: Grund. der Math. Wissen. 313, Springer-Verlag (1997). [20] Norris (J.R.), Twisted Sheets. J.Funct. Anal., 132(1995) 273-334. [21] Rockner(M.) and Zhang(T.S.) , Finite dimensional approEimation of diffusion processes o n infinite dimensional spaces. Stoch. Stoch.and Rep., 57, 37-55(1996). [22] Yosida ( K . ) : Functional Analysis, Sixth Edition, Springer-Verlag, 1980.

Essential spectrum on Riemannian manifolds * K. D. Elworthy Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. email: [email protected] Feng-Yu Wang Department of Mathematics, Beijing Normal University, Beijing 100875, China. email: [email protected]

Abstract By using a measure transformation method, the essential spectrum of the Laplacian in a noncompact Riemannian manifold is studied. The principle result given includes the corresponding main theorems in both Kumura [16] and Donnelly [9]. The essential spectrum of the Laplacian on one-forms is also studied.

Key Words. Essential spectrum, Laplace operator, Riemannian manifold.

1. Introduction Let M be a complete connected Riemannian manifold of dimension d 2 2. Denote by A the Laplace-Beltrami operator. The purpose of this note is t o estimate cess(-A), the essential spectrum of -A on L 2 ( p ) ,whcre /I is the Riemannian volume measure. Let 1). denote the LP(p)-norm, p >_ 1.

\Ip

*Research supported in part by NNSFC(10121101, 10025105),TRAPOYT and the 973-Project. 151

152

Let us first recall some previous results in the literature. In 1981, Donnelly [8] proved uess(-A) = [(d - 1)2k2/4,00) provided M is a Hadamard manifold with sectional curvatures approaching - k 2 a t infinity. In 1994, Li [18] proved ness(-A) = [O,co) if M has nonnegative Ricci curvatures and possesses a pole, see also Escobar-Freire [14] for the c a e that M has nonnegative sectional curvatures. In 1997, Kumura presented the following result [16, Theorem 1.21: let T be the distance function from a pole (more generally, a regular domain such that the outward-pointing normal exponential map induces a diffeomorphism), then uess(-A) = [c2/4,co) provided

lim sup lAr

- cJ =

0.

n--*co r>n

Kumura also proved that this result recovers all those mentioned above. Moreover, [9, Theorem 2.41 says that uess(-A) = [0, co) provided there exists a C2-exhaustion function y satisfying

1 (Ay)2]VOl(dz) = 0, (i) lim t+covol(y 5 t ) [(IW(ii) I v y / + lAyl is bounded and there exist c > 1 and p E M such that c-ldist(p, z) 5 y(z) 5 cdist(p, z) holds outside a compact set.

Sblt

+

As is pointed out in [7] these assumptions preclude exponential volume growth of the manifold. Essentially they therefore refer to a different class of manifolds from those covered by Kumura. We refer t o [a, 31 for constructions of the function y under certain curvature conditions. See [lo] for further study concerning the absence of point spectrum and singular continuous spectrum. This paper is derived from the preprint 1131 which was stimulated by Kumura’s work. Our main result, Theorem 1.1, improves and simultaneously generalizes the above results by Kumura [16] and Donnelly [9]. Some variations are given in Section 3 and we also briefly consider the one form case (see section 4). For a description of the essential spectrum of manifolds with ’multicylindrical ends’ obtained by very different techniques see 1171. See [lo] for further study concerning the absence of point spectrum and singular continuous spectrum. Recently, the essential spectrum was also studied by using functional inequalities, see [22, 231 and references within. Let y E C ( M ) be a function satisfying (1) y is unbounded above and is C2-smooth in the domain {y > R } for

153

some R > 0. (2) p ( ( m 0 < y < n } ) < 03 for some mo and any n > mo, where p is the Riemannian volume element.

Theorem 1.1. For a n y t > 0 and c E R, let Bt = {x E A4 : y(x) < t } ,dpc = epCYdp and Uc(s,t ) = pc(Bt \ B,) f o r t 2 s. If there exists c such that

and lim max Uc(mo,t ) ,Uc(t,m)-'}e-''

t-im

{

=0

f o r any

E

> 0,

(1.3)

then uess(-A) 3 [ c 2 / 4 ,00). Especially, i f (1.2) and (1.3)hold f o r c then uess(-A) = [0,m).

=

0,

Moreover, i f in addition (3) {y 5 n } is compact f o r a n y n > 0 and JVyJ4 1 uniformly as

y 4 00.

+5

Then uess(-A - :AT) = [0,cm) f o r any C2-smooth function that 7 = y outside a compact set.

7 such

To see that Theorem 1.1 recovers the above Kumura's result, let y := r be the distance function from a regular domain which is smooth when r > 0. Let Y be the Lebesgue measure induced by the outward-pointing normal exponential map, i.e. under the polar coordinates ( r , 0. Then, for any s > 0,

This implies (1.a). Finally, to present the exact estimate of gess(-A) for c # 0, we assume that Pt := eta is ultracontractive [7]: there exists T > 0 such that lI~Tll2--tm =: C ( T )

< 03,

(1.4)

where 11. )12+m is the operator norm from L 2 ( p )to L"(p). In particular, if either the Ricci curvature of M is bounded from below and the injectivity radius is positive (see [21], or [6, Corollary 2.4.31 and [19]),or the injectivity radius is infinite (see [l]or [ 5 ] ) ,then there holds the Nash inequality

~ ( f ~ )I~c(p(ivfi2) + ~ / ~ + P ( ~ ~ ) ) P ( I ~ If) E~ c / ~ ,a ~ ) for some constant C

> 0, and hence llPT112'03

(see e.g. [6, Corollary 2.4.71)

I C'(1 + T d l 4 )

for some C' > 0 and all T > 0. Thus, the above geometric conditions imply the ultracontractivity. See [6, 4, 221 for more descriptions of the ultracontractivity using functional inequalities. See [2O] for explicit criteria for the ultracontractivity on Riemannian manifolds.

Corollary 1.2. Suppose that (1.2) and (1.3) hold with y satisfying (1) and (3), and (1.4) holds for some T > 0. If there exists f E L P ( p ) for some p E [l,a)such that Ay - f 4c uniformly as y + 00, then gess(-A) = 2 [p, 2. Proofs Proof of Theorem 2 . 1 . We prove the two assertions respectively. a) By (1.3), there exist ,s cm and t, t 00 such that

156

+ +

+

Let X 2 c2/4 be fixed. For any t > s 1 > (Y := max{m 1 , 2 R}, choose h E C"(R) such that 0 I h I l l h ~ [ s , t = - ~ l~l ~ l ~ - m , = s -hI[t+,) l~ =0 and Ilh'llwl Jlh"ll, 5 co for some constant co independent of s,t. Next, let A, = and

,/-

fl(s) = epcSI2sin(Xcs),

f2(s)

= e-cs/2 cos(Xcs), s

2 0.

Finally, let

+

Uc(s~ -

By (2.1) and (2.2) we have

/

{(Ay - c)'

l)) Bt\B,-I

+(

- 1)'}dpC.

157

Therefore, by (2.3) we obtain

= lim

{ (A7 - .I2

lim

+ ( l b 1 2 - 1I2}dPc

= 0.

(2.4) This implies that X E .(-A) and hence .(-A) 3 [$-,a). b) Let L = A-cVy. Then L is symmetric with respective t o p c := e-"p and we shall always consider it as an operator on the space L2(pc). For X 2 0 so that 7 = y on

{Y 2 SIT (L

+

- c2/4)tjs,t = eCYl2{ (A

+ X)g,,t + c(Ay - c)gS,,/2 + ( IVyI2 - l)c2/4}.

Then

~ . a) and condition 3) we obtain X - $- E a(-L). since ( i j s , t (5 I B ~ \ B , -By Hence a(-L) = [0,cm) since -L 2 0. Finally, noting that

(A we have cess(-A

aess(-A

+ cA7/2) fe-cr12

= e-cy12Lf ,

f

E C2(M),

(2.5)

+ s > R, let X,,t be the first Dirichlet eigenvalue of -A in B,,t =: Bt\B,, with the normalized eigenfunction 4s,t2 0. We extend +s,t E C ( M ) by taking q 5 , , t l ~ : , , 0. From ( 2 . 5 ) we know that -A $lVrlZ- ;AT 2 0 in L 2 ( p ) . Then

=

+

where

as s 4 00 by assumption. Let ( P > t )~ >be O the Dirichlet semigroup generated by A on B,,t. By (1.4) and noting that A4,,t = -Xs,tds,t on B,,t, we obtain

Il4s,tllm

Therefore, for any p

If f

5 C(T)eXs3".

2 1,

E LP(p), then by letting s -+

00

we arrive at

A0

2 .:

0

3. Extensions Recall that in Theorem 1.1, we consider the reference measures dpc = e-C7dp. In general, we may replace p, by any other measure with certain condition on the volume growth. This is the main idea of this section.

159

For positive g E C 2 ( M ) ,let dp, = gdp. Then we have the following result.

Theorem 3.1. Suppose that y satisfies (1) and (2). Let U,(s,t) B,) for t 2 s and V, = g-1/2Ag1/2. If

= p,(Bt

\

and

{

lim max U,(mo, t ) ,U,(t, c o ) - ' } . F E t = 0

t4Cc

then a(-A

for any E > 0,

(3.2)

+ V,) = [0,co).

Proof. Let L = A + Vlogg. We have (-A + Vg)g1/2f = -g1/2Lf for f E C 2 ( M )and using this conjugacy we can, and will, consider L as a selfadjoint operator on L 2 ( p g ) .Then a(-A V,) = a(-L). Hence, it suffices to prove o(-L) 2 [0, CO) since -L 2 0. Actually, for any t > s 1 > a , let h be as in the proof of Theorem 1.1. For X 2 0, set

+

gs,t = h(y) cos(Xy) or

+

h(y) sin(Xy)

such that JJg,,tJ/$(pg)2 $U,(s,t - 1). Noting that gy,t [sit - 11, we have

+ X2gs,t = 0 on

for some .(A) > 0. Then the remainder of the proof is similar to the proof of Theorem 1.1. 0

Remark. Let us consider Rd with the Riemannian metric g(a,,aj) = g i j ,

i,j

=

l , . . .,d.

Let g = (detgij)1/2. Then the corresponding Riemannian volume measure is p(dx) = g ( z ) d s and the associated Laplacian operator writes

160 d

C (8igij8j + (gij8ilogg)8j),

A, =

i,j=l

where (gij) is the inverse matrix of (gij). We have d

L :=

C 8igij8j = A,

- V,logg,

ij=l

where V, is the gradient operator induced by g . Then the proof of Theorem 3.1 shows that a ( - L ) = [0,CQ) on L2(dz) under the conditions of Theorem 3.1. From now on, we assume that r is the distance function from a regular domain such that the outward-pointing normal exponential map induces a diffeomorphism. Let u be the Lebesgue measure induced by the exponential map and let 0 = dp/dv which is well defined out of the domain. Then (3.1) and (3.2) hold with the choice g = 0 - l . Therefore, by Theorem 3.1 we always have

ness(-A

+ 01/2AO-1/2)= [0,

CQ),

(3.3)

Then we can describe uess(-A) by studying the potential & =: 01/2AO-1/2. We emphasize this potential because it plays an important role in the heat kernel formula due t o Elworthy and Truman, e.g. see [12].

Corollary 3.2. 1) We have gess(-A) = [-c, C Q ) if limn+oo suprzn IVo CI = 0. 2 ) We have oess(-A) 2 [-c, C Q ) provided

-

(3.4)

Especially, ness(-A) = [0,C Q ) i f (3.4) holds with c = 0. To prove this result, we need the following lemma.

Lemma 3.3. For any X 2 0 and n 2 1, there exasts gn E CF(B,) which depends only on r , such that JJgnJJLz(,) = 1,JJgnlJoo 5 c(X)nPdl2for some c(X) > 0 and

161

where Lr = $

(d-l)a + r.

Proof. Choose h E Cm(R) such that 0 5 h 5 1,hl(-m,-l] = 1,hlp,m) Take

= 0.

where c, = llh(r - n ) cos(v'%-IILz(v) 2 c(X)-'nd/' for some c(X) > 0. Therefore, there exists a constant C

> 0 such that

+

Proof of C o r o l l a y 3.2. We need only to prove 2). Let H = -A Ve. For any X 2 0, let {gn}n22 be chosen as in Lemma 3.3. Noting that H Sn - - ,5-1/2 (-L,)g, for ijn = Q-1/2gn, we have

1I-A

+ c)ijn - w2 F II(b- .)9,112 + ll(H

-

X)ijn112

+ I I ( b +X)gnllu(v) i c(X)n-d/211(Ve - C)1BnllL2(v) + I I ( L + x)gnlla

=

which goes to zero as n [O,

.I..

II(Ve - c)g,llu(v)

-+ 00

by (1.1) and (3.5). Hence, a(-A

+ c) 3 0

Corollary 3.4. W e have aess(-A) = [-c,00) if there exists f such that Vtheta - f 4 c uniformly as r + 00 and one of the following holds: 1) (1.4) and (3.4) hold and f E L p ( p ) for some p E [I,co). 2) f E L P ( p ) for some p > 2 and the following Sobolev inequality holds: 2

119112p/(p-2) < c(P(lvg12) + P(S2))l

9 E C,-(M).

(34

Noting that the sufficiency of condition 1) in Corollary 3.4 follows immediately from Corollary 3.2 and the proof of Corollary 1.2, we need only to prove the sufficiency of condition 2). We remark that (3.6) holds for p E (d, m) n (2, co) provided either the injectivity radius is infinite or it is positive but the Ricci curvature is bounded below, see references mentioned after (1.4).

162

Proof of Corollary 3.4. It suffices to prove the result using condition 2). Suppose that f E LP(p) for some p > 2. Let H = -A+f+c, then uess(H) = aess(-A Ve) = [0, m) since Vo - f - c 0 uniformly as T -+ 00. So, for any X 2 0, we may choose {gn}nzl such that 11gn112 = l , g n E C p ( B i ) and

+

--f

(3.7)

+

since f E L P ( p ) . By combining this with (3.9), we obtain X E a(-A c). Hence uess(-A c) 3 [0, m) = a ( H ) . Similarly, we can prove the inverse inclusion.

+

Remark. The proof of Corollary 3.4 shows that the second part in this corollary is also true with Ve replaced by V, provided (3.1) and (3.2) hold. 4. The essential spectrum on one-forms

Let Ap = -(db+Sd) be the Laplacian on hP(M) for 1 5 p 5 d. Escobar and Freire [15]proved that u e ss(-A p )= [0,m) if the radial sectional curvature,

163

the curvature tensor and the first derivative of the curvature tensor decay to zero at certain speeds. See also [17] for results concerning manifolds with ‘multicylindrical ends’. Here, we try to extend Theorem 1.1 to the one-form case.

Theorem 4.1. Suppose that y satisfies (1) and (3) and let U, be as in Theorem 1.1. If there exists c such that (1.3) holds and

+

$ - ZA7) = [0,co)for C2then cess(-Al) 3 [C2x , m ) and cess(-Al smooth 7 such that 7 = y outside a compact set. Proof. For X > $, let

f1

and

f2

be defined as in section 2. We have

+ f;(s)2 = Xe-”,

s

2 0.

(4.2)

Next, let h be chosen as in section 2 for t > s + l > a with h”’ also uniformly bounded. By (4.2) and condition ( 3 ) , we may take either gs,t = h ( y )fl(y) or g,,, = h ( y )f2(y) such that

llvgs,tll;

X

2 q U , ( s , t - l),

Now, take w,,t = dg,,,, we have in Bt-l

s

>> 1.

\ B,,

By (1.3), (4.1) and the proof of Theorem 1.1a), we prove the first assertion. As for the second assertion, let L = A, - c V v y , we have

( - A l + -IV7l2 4 C2

C

- -A-;)e-C4/2w = -e-cry/2Lw,

2

w E A1(M).

Hence, we need only to show that uess(-L) 3 [O,m) since -L 2 0. Let = ecT/2ws,t,we have

164

Then the remainder of the proof is similar to the proof of Theoreml.1 b.

0 Corollary 4.2. Suppose that r is the distance function f r o m a regular domain which is smooth in { r > 0 ) . W e have oess(-Al) = [$,co) provided lAr - cI

+ IVArl + 0

uniformly as r

+ 00.

Acknowledgment. The authors would like t o thank the referee for his corrections.

References Chavel, I., Eigenvalues in Riemannian Geometry, New York: Academic Press, 1984. Cheeger, J. and Colding T., Lower bounds o n Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144(1996), 189237. Colding, T. and Minicozzi, W., Large scale behavior of kernels of Schrodinger operators, Amer. J. Math. 199(1997), 1355-1398. Coulhon, T., Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141(1996), 510-539. Croke, C.B., Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ec. Norm. Super. 13(1980), 419-435. Davies, E.B., Heat Kernels and Spectral Theory, Cambridge University Press, 1989. Davies, E. B. and Simon, B., Ultracontractivity and the heat kernel for Schrodinger operators and Dirichlet Laplacians, J. Funct. Anal. 59(1984), 335-395. Donnelly, H., O n the essential spectrum of a complete Riemannian manifold, Topology 20(1981), 1-14. Donnelly, H., Exhaustion functions and the spectrum of Riemannian manifolds, Indiana Univ. Math. J. 46(1997), 505-527. Donnelly, H., Spectrum of the Laplacian o n asymptotically Euclidean spaces, Mich. Math. J. 46(1999), 101-111. Donnelly, H. and Li, P., Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J. 46(1979), 497-503. Elworthy, K. D., Geometric aspects of diffusions o n manifolds, Lecture Notes in Math. 1362(1988), 276-425.

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[13] Elworthy, K. D. and Wang, F. Y., O n the essential spectrum of the Laplacian o n Riemannian Manifolds, Warwick Preprint:30/1997. [14] Escobar J.F. and Freire, A., The spectrum o f t h e Laplacian of manifolds of positive curvature, Duke Math. J . 65(1992), 1-21. [15] Escobar, J. F. and Freire, A., The difjerential t o m spectrum of manifolds of positive curvature, Duke Math. J. 69(1993), 1-41. [16] Kumura, H., O n the essential spectrum of the Laplacian o n complete manifolds, J . Math. SOC.Japan 49(1997), 1-14. [17] Lauter, R. and Nistor, V. O n spectra of geometric operators o n open manifolds and diflerentiable groupoids, Electronic research announcements of AMS 7(2001), 45-53. [18] Li, J., Spectrum of the Laplacian on a complete Riemannian manifold with non-negative Ricci curvature which possesses a pole, J . Math. SOC. Japan 46(1994), 213-216. [19] Li, P. and Yau, S.T., O n the parabolic heat kernel of the Schrodinger operator, Act Math. 156(1986), 153-201. [20] Rockner, M. and Wang, F.Y., Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds, t o appear in Forum Math. [all Rosenberg, S. and Yang, D., Bounds o n the fundamental group of a manifold with almost nonnegative Ricci curvature, J . Math. SOC.Japan 46(1994), 267-287. [22] Wang, F. Y., Functional inequalities for empty essential spectrum, J. Funct. Anal. 170(2000), 219-245. [23] Wang, F. Y., Functional inequalities and spectrum estimates: the infinite measure case, J. Funct. Anal. 194(2002), 288-310.

Levy process on real Lie algebras Uwe Franz Institut fur Mathematik und Informatik Ernst-Moritz-Arndt-Universitat Greifswald Friedrich-Ludwig-Jahn-Str. 15 a D-17487 Greifswald, Germany Email: [email protected] http: // hyperwave .math-inf.uni-greifswald.de/algebra/franz

Abstract The definition and the basic theory of LBvy processes on real Lie algebras is summarized and several examples are presented. It is also shown how KMS or temperature states can be constructed for such processes.

1. Introduction Lkvy processes on real Lie algebras form an interesting special case of Lkvy processes on involutive bialgebras. They have already be studied under the name factorizable representations of current algebras in the sixties and seventies, see [lo] for a historical survey and for references. They are at the origin of the theory of quantum stochastic differential calculus. In this paper we give their definition and summarize their main properties, cf. Section 2. In Section 3 we consider two examples. The first is a Lkvy process on the oscillator Lie algebra, which is closely related to Hudson-Parthasarathy quantum stochastic calculus. The second example is a LQvyprocess on the Lie algebra d ( 2 , IR), which appeared recently in an attempt to define a quantum stochastic calculus for the squares of creation and annihilation via a renormalization, cf. [3, 9, 21. In Section 4 we present some new material. We show how one can construct KMS or temperature states w.r.t. a given automorphism of a real Lie algebra. Actually, since the proofs of fundamental results in Lemma 4.1 and Proposition 4.2 use bialgebra techniques, the last section is written for Lkvy processes on involutive bialgebras in general. Our main result 166

167

is that the law of a Lkvy process satisfies a KMS condition if and only if its generator does. Therefore it is sufficient to find generators satisfying a KMS condition. A fairly general method for finding such generators can be obtained by generalizing a construction due t o Amosov, cf. [l],see Theorem 4.5. Finally, we construct an example on the Lie algebra d ( 2 , R). 2. LQvyprocesses on real Lie algebras In this section we give the basic definitions and properties of Lkvy processes on real Lie algebras, see also [a]. This is a special case of the theory of L6vy processes on involutive bialgebras, for more detailed accounts on these processes see [8], [6, Chapter VII], [5]. For a list of references on factorizable representations of current groups and algebras and a historical survey, cf. [lo, Section 51. Let D be a complex pre-Hilbert space, then we denote by L ( D ) the algebra of linear operators on D having an adjoint defined everywhere on D , i.e. L ( D ) = {A : D -+ D linear1 there exists a linear operator A* : D 3 D s.t. ( 5 ,Ay) = (A*%,y) for all b'x, y E D } . By .LAH(D) we mean the anti-hermitian linear operators on D,

L A H ( D )= { A : D

-+

D linearI(s,Ay) = -(As, y) for all z, y

E

D}.

Definition 2.1. Let g be a Lie algebra over R,D be a pre-Hilbert space, and R E D a unit vector. We call a family ( j s t : . L A H ( D ) ) , , ~of, ~ ~ representations of g a L&vy process o n g over D (with respect to R), if the -+

following conditions are satisfied. (1) (Increment property) We have

+j t U ( X ) = j S U ( X ) for all 0 5 s 5 t 5 u and all X E g. (2) (Boson independence) We have [ j s t ( X )jyftl , ( Y ) ]= 0 for all X,Y E g, 0 5 s 5 t 5 s' 5 t' and

(a,j S , t ,

( X d k l * * . j s n t n( X n ) k " Q = ( Q , j S l t l (Xl)klQ2). . . (0,jY,t,(X,)"-R) foralln,kl, ..., kn E N , O ~ S ~ L ~ ~ _ < S ~ L .. . .., X I ,~E, , X ~ , 0.

(3) (Stationarity) For all n E

N and all X

E g, the moments

mn(X;s, t ) = (fl,jSt(X)"R) depend only on the difference t

-

s.

168

(4) (Weak continuity) We have limt\s(R,j,t(X)"R) = 0 for all n E and all X E g.

N

Such a process extends to a family of *-representations of the complexification g@= g @ ig with the involution ( X iY)*= -X iY for X I Y E g by setting jst(X ZY) = j , t ( X ) zjst(Y).We denote by U ( g ) the universal enveloping algebra of gc and by Uo(g)the non-unital subalgebra of U generated by g. If X I , . . . , x d is a basis of g, then

+

+

+

+

{X,"' . . ~ X ~ d ( n. .l,,n .d E N,n1+ ... + n d 2 1) is a basis of Uo(g).Furthermore, we extend the involution on gc as an anti-linear anti-homomorphism to U ( g ) and Uo(g). If ( j s t ) o l s l t is a LQvy process on g, then it extends to a family of *representations of U ( g ) , and the functionals pt = (R,jot(.)R) : U ( g ) 4 C are actually states. Furthermore, they are differentiable w.r.t. t and

defines a positive hermitian linear functional on Uo(g).In fact one can prove that the family (pt)t>o - is a convolution semigroup on U ( g ) whose generator is L. The functional L is also called the generator of the process. Let ( j $ ) : g -+ L A H ( D ( ~ ) ) ) and ~ ~ ~ (~j (~ 2 ): g 4 L A H ( D ( ~ ) ) ) ~ be< ~ < ~ two LQvyprocesses on g with respect to the state vectors and R(2), resp. We call them equivalent, if all their moments agree, i.e. if

(dl) ,j:;) (X)"(1) ) = (R(2),j p ( X )W)), for all k E agree,

N,0 5 s 5 t , X

E g. This implies that all joint moments also

x 1)

(0(1), 3.(I) s1t1( =(R(2),jg;'

kl ,

(X,)kl

. .+;\n(xl)&~(l)) (Xn)knR(2)),

f o r a l l n E N , O _ < s lst1 < s 2 < . . . ~ t n , u l, . . . , u , E U ( ~ ) . By a GNS-type construction, one can associate to every generator a Schiirmann triple.

Definition 2.2. A Schurmann triple on g is a triple ( p , 77, $), where p : g + CAH(D)is a representation on some pre-Hilbert space D , i.e. P( [ X ,YI) = P ( X ) P ( Y )- P ( Y ) P ( X ) and

d X ) *= - d X )

for all X , Y E g, 77 : g + D is a pl-cocycle, i.e. it satisfies

rl(KYI) = P ( X ) V ( Y )- P ( X ) d Y ) ,

169

for all X , Y E g, and $ : g 4 C is a linear functional with imaginary values ) the 2-coboundary of such that the bilinear map ( X , Y ) H ( q ( X ) , q ( Y ) is $ (w.r.t. the trivial representation), i.e.

+ ( [ X ,YI) = (rl(Y),rl(X))- (rl(X),rl(Y)) for all X, Y E g. The functional $ in the Schurmann triple associated t o a generator

L : Uo(g)-+C is the restriction of the generator L to g. Conversely, given a Schurmann triple ( p , rl, $), we can reconstruct a generator L : Uo(g)4 CC from it by setting

L(X1 . . . X n ) =

{

$(Xl) -(rl(X1), r l ( X 2 ) )

-(

for 72 = 1, for n = 2, rl(Xl),P(X2>...P(X,-l)rl(X,)) for 72 > 2,

for X I , . . . ,X , E g. We will now see how a LQvy processes can be reconstructed from its Schurmann triple. Let ( p , 7, $) be a Schurmann triple on g, acting on a pre-Hilbert space D. We can define a Lkvy process on the symmetric Fock space l?(L2(R+,0 ) )= @ = ,: L2 (R+, D)@" by setting

j s t ( X ) = Ast ( P W )

+ A:t (rl(X))- Ast (rl(X1)+ $ ( X ) ( t- s)id,

(2.1)

for X E g, where Ast, A:t, A,t denote the conservation, creation, and annihilation processes on r(L2(R+,D)), cf. [7, 61. It is straightforward t o check that we have

[ j s t ( X ) , j s t ( Y )= ] j s t ( [ X , Y ] ) , and

j s t ( X ) *= - j s t ( X )

for all 0 5 s 5 t , X , Y E g, and that the moments of js, satisfy the stationarity and the continuity property. Furthermore, using

~ ( L ~ ( R0+),) r(L2([0,t[,0)) r ( ~ ~+m[, ( [ t0 , )) and the fact that R E R 8 R holds for the vacuum vector w.r.t. to this factorization, one can show that the increments of ( j s t ) o l s < t are boson independent. By the universal property, the family

extends to a unique family of unital *-representations of U ( g ) , which we denote again by ( j s t ) o S s < t .

170

The following theorem shows that the correspondence between (equivalence classes of) LQvy processes and Schurmann triples is one-to-one and that the representation (2.1) is universal.

Theorem 2.1 ( [ 8 ] ) . Two Le'vy processes o n g are equivalent if and only i f their Schurmann triples are unitarily equivalent o n the subspace p ( U ( g ) ) v ( g ) . A Le'vy process (kst)O A 1 > A 2 > . . . be the eigenvalues of the operator A$ and let $1, $2, . . . be corresponding eigenfunctions. (a) It holds that Z(&) = k f 1, k E N. (b) Let ux, X 5 0, be the family of functions introduced an L e m m a 4.1. W e have

This yields the following result, see [5]

Theorem 4.1. Let X 5 0 and let px E D(AP) \ (0) satisfy the equation = xpx. (a) If A1 < X 5 0 then we have 0 5 Z(pA) 5 1. (b) We have

Apx

4.2. The asymptotic distribution of the zeros i n the self-similar case In this subsection, we restrict ourselves to self-similar measures p which has been introduced in Subsection 3.1. This means, p is the unique Bore1 probability measure with respect t o a vector of weights Q = (el,. . . , Q M ) and a finite family of affine contractions S1,. . . , SM mapping from [0,1] t o [0,1] with ratios r i such that (3.1) is satisfied, i.e. the images of two different Si are disjoint or just-touching. Moreover, the support of p is given by the compact set L, which is the unique self-similar set with respect t o the family S = (S1,. . . , S M } . For such measures p , we determine the asymptotic distribution of the zeros of the eigenfunctions of the corresponding operator A;. This is only possible in the non-arithmetic case (see Theorem 3.2). Note that the results of this

195

subsection have been obtained in [5],in the particular case that p is a selfsimilar measure which is supported on the classical Cantor set introduced in Remark 3.1. In order t o formulate the next lemma, let us first introduce the notion of the so-called Code space. Let W = (1, 2 , . . . , M}N be the set of all infinite words consisting of the letters 1 , 2 , .. . , M . Furthermore, we define for any k E N U (0) the space W k := (1,2,. . . , M}’” of the words of length k, setting WO:= (0) where (4 is just the empty word. For any w E W k , i.e., w = (w1,. . . wk) for some wi E (1,2,. . . ,M } , i = 1,.. . , k , we introduce the notations S, := S,, 0 . . . o S,,, ew := p,, . . . . . pw, and r , := r,, . . . . .r,,. Finally, we set Sn :=id, ~0 := 1 and 7-0 := 1. The following lemma provides a renormalization property of the operator A” under the particular choice of p = p(S, @).

Lemma 4.4. Let X < 0 and let p be a solution of APp W u ( 0 ) and w E W k , define (Put

=

X p . For k E

sw .

:= ‘ p l S W ( [ O , lO] )

Then we have App,

=

ewrwX . p w ,

wE

Wk,

k E

N u (0).

Proof. The lemma will be proved by induction. The case k = 0 is trivial because 90 = p. Fix k E N U (0) and let Ap, = ewrwX .p,, w E W k . Take 6 E W ~ +arbitrary, I so we have 6 = (w,wk+l)for some w E Wk,wk+l E (1,2,. . . , M ) . Having in mind Remark 2.2, it follows that CpG(X)

= cpw(~w,+,(~)) - (PW(sWlc+l (O)) f ( p W ) ’ ( s W k + l

(o))[SWk+l(z>

- sWk+l

Hence (using again Remark 2.2), we have shown that App6 = pGr-6

96,

In order t o formulate the following theorem, define ci := (piri)y, i = 1 , 2 , .. . ,M ,

(‘11

196

where y is the spectral exponent, given in Theorem 3.1. By the definition of y, (T = ((TI,. . . , C T M )is a vector of weights. Let v be the unique Bore1 probability measure with respect t o the family S and the vector CT. Obviously, we have supp v = L. As in Proposition 4.3, let $, be an eigenfunction corresponding t o the eigenvalue A, of A;, n E N. For any set A C [0,1],let &(A) denote the number of all zeros of $, belonging to A. Furthermore, define

z,(x) :=

zn([o,51) , n+l

5 €

[O, 11;

note that z, : [0,1] 4 [0, I] is independent of the choice of the eigenfunction $, to the eigenvalue A., Finally, let denote the probability measure on ([0,1],B([0,1])) the distribution function of which is z,, n E N.

0 real, . complex analytic extension of the Euclidean scalar product a . b = C",=, anbn, a,b E Cd+', @ semi-direct product, E ( d ) Euclidean group on Wd,L(d) the Lorentz group on Rd, L,(d) its complexification, ( 2 - z)2 = ( z - z) . ( z - z) where the scalar product is as above but in Cd.

~

215

imation of Gl12 for 6 > 0 small. The following Lemma shows that such UV-regularized kernels fit into our scheme:

Lemma 5.4. The kernels G, have holomorphic extension G: to C” such that GC(z,x)= G:(z-x), x E R d , z E Cd fulfills the requirements (I)-(IV).

Proof. (I) Follows from

To check (11), define G“(z,x) = JRdg,(z - x - x’)G1/~(x’)dx’, 2 E R d l z E Cd. For K 2 Cd compact, let gK(z) be defined as in Condition (11). We have g K ( x ) Jid supzEKIgE(z - x - x’)lGl/2(z’)dx’. Obviously, supzEKIgE(z- x)I E L1(Rd,dz) and therefore the r.h.s. of this inequality is in L1(Rd,dx) as a convolution of Lebesgue integrable Clearly, gK is continuous and functions. Thus also gK E L1(Rd,dx). gK(z) 5 SUPzEK+idIg(z)lIIG1/211Ll(id,dz)< m v x E hencegK E c b ( x ) . (111): Taking into account the definition of G “ ( z , x ) and the fact that gE(z)is holomorphic on Cd in combination with the estimate in (II), one gets that the assumptions of Morera’s theorem 2.5 are fulfilled. Holomorphy of G“ in the first argument follows. (IV) Let cy E O(d) and a E Rd. Then


-~lcl;l,

223

a.

In the sequel Act,q) will be abbreviated as The following theorem gives the sufficient and necessary conditions under which a two-variable complex function on E x E is the symbol of generalized operator, which is essentially due to Potth~ff-Streit[~I, Obata[1013[11]and Kondratiev-Leukert-Potthoff et a1[81.

Theorem 2.1. Let G be a complex function o n E x E satisfying that (i) (Analyticity) f o r any E l , 6 2 , 7 7 1 , ~E~El G(E1 SEZ, ql t772) has an analytic continuation to (s, t ) E a'z; (ii) there exist constants C,K > 0 , p E R? such that

+

+

a( 0 , such that KE(0)t

IB2u(x,t)10I Ce

(3.6)

holds for V ( 2 ,t ) E R3 x R+ Proof. Let G =

(f :),

@ ( t )=

(:[:i)

E

'FIB be the solution of

eq.(5). Then

Since AG

= GA, the

unitarity of e-itA implies that

+

IB2u(t)I0I IIG@o~~B IIGe-i(t-")AJ IIBds = IIG@o~IBX2 s," IBu2(S)G(S)lodS I lIG@OlIB A2KJ: IBu(S)IgIB2u(S)10dS,

+ +

where the last inequality is obtained by Lemma 3.4. It is obvious that IBu(s)Ii 5 2E(O),then by Gronwall inequality we have KE(0)t

IB2u(t)105 IIG@OllBe

229

Proof. By taking S-transforms of generalized operators $(x), ~ ( xwe) have

=

where

F

i

-&(Fr,

- F-l[)(x),

Jz

is the Fourier transformation. Then

IB&J

=

+

&IB+(.Fr, F - l J ) I o

I K(IB'(Fr,)lo + IB+(.F-lt)lo)= I r ' ( / w ~ r , l o+ I w i J l o ) I K(IAar,lo + IAatlo) = K(l~l;+ IEI;). Therefore (i) is obtained. By the same way we have (ii), (iii) By Lemma 3.4 and Schwartz inequality we have

I sR3(h4(x)dx~= I I ~ 161~iid116" I II :~-1 ~ 4i~i~(hi; 5 KIB$I: 5 K(IEI;'/4 + IVI$4).

Proof of Theorem 3.2. Let E,r, E E be fixed, by taking S-transform of eq. (3) we get a 4-dimensional space-time classical non-linear wave equation

$d(t, x) - B2d(t,).

=

;xld(t,

x)12d(t,).

d(0,4 = 4(x) &&o, x) = ii(x).

(3.7)

Since $(x), $5) are Fourier transformations and inverse transformations of

E, r, E E , there exists a unique global solution {$(t,x),(t,x) E [0, +m) x R 3 } of eq.(7)(cf. [16], [18]). By Theorem 3.6 we know that

IB2(h(x)lo5

(lB2@

+ IB@)+eKE(O)t,

where

+

{IBd12

E ( 0 )= 1 2

L

+ pb4}d3x x-

3

is the conserved energy of eq.(7). By the non-decreasing property of countably Hilbertian norms and the Lemmas above, the following inequality 1 ~ 2 d ( t)lo x , 5 ~~eK2(IEI~/4+Il7I~/4)t

230

holds. Using Sobolev embedding theorem we have

I$(t,z)l 5 m y I4(t,z)l 5 I P 4 ( t ,z ) l o 5

~le~z(IE(~/4+1~1~/4)~.

xER

The same arguments as Theorem 3.1 demonstrate that we obtain a gener1 1 alized operator-valued function 4(t,z) E C ( ( E ) z (E);') , of ( t , z) which is referred t o as (4): quantum field.

3.3. The Dynamical Properties of Interacting Quantum Field Since the dynamical properties of the two quantum fields are very similar, we only give some properties of (4); quantum field.

It is natural t o define the renomalized Hamiltonian of eq.(3) by

a4

's

H ( t )= 2

x

{: IB4(t,z)(2: + : I-(t,z)(2 : +- : 44(t,z) :}d3z. at 2 1

1

Theorem 3.8. H ( t ) E C ( ( E ) z ,( E ) i 3 )is independent o f t , i.e. H ( t ) = H ( 0 ) fort 2 0. Theorem 3.9. Let P ( t ) = -SO& : V,4d3x be the renormalized moa =, a -)a are gradient operators. menta, where Vt = and V, = (=,

&

Then P ( t ) = P(0) and each component o f P ( t ) is in L ( ( E ) &(E),'). Proof. The S-transform of P ( t ) is

-

v&, z) . V&t,4 d 3 z .

P ( t )= L

3

Actually, 6 ( t )is exactly the momenta for classical field equations and hence a conserved quantity by Noether Theorem, i.e.,

-

-

P ( t ) = P(0) = -

Z(IC)V&(z)d3z, L

3

By Theorem 3.8, the Hamiltonian H ( t ) of eq.(3) can be written as

H

+

= H ( 0 )= Hfree

(3.8)

231

where "

4

a* a*P3 ap4 + 6e-i(Pl+PZ-P3-P4)."a* a* a a Pl PZ P 3 P4 + 4 e - i ( ~ l - ~ ~ - ~ ~ - ~ 4 )a. x ,a* a p1 PZ P3 + e-i(-Pl-PZ-P3-P4)~x a,, a,, a,, a,, V P l dP2 dP3dP4. + 4e-i(Pl+PZ+P3-P4).xa*

Pl

P2

P4

Since the second term of the right hand side of eq.(8) only a quadratic form on (E)hI2,although Hisfree is essential self-adjoint on Fock space, H may be not a proper operator. For this reason, let X = X(z) E S(R3)be a space cut-off. Thus H is replaced by

Because ( L ,:) is a commutative nuclear algebra, once involving the CCR of quantum field theory, we must consider in the sense of operator-valued distributions. Suppose that f , g E S ( R 3 ) and , put

where the last two equations should be understood in the sense of operatorvalued distributions.

232

Proof. Eqs.(9) and (10) can be verified by simple calculations. Since the proof of [Ho,$(f)]= - i 7 r ( f ) is not a complicated business, e q . ( l l ) is reduced to prove that [: $‘(A) :, $(f)]= 0. By the definition of Wick products we have

: 4 4 ( X ) :=

holds. Hence the proof is complete.

References [l] J . C. Baez, Wick products of the Free Bose field, J. Func. Anal., 86 (1989), 211-225. [a] J . Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer, Berlin/Heidelberg/New York, 1981. M. Grothaus and L. Streit, Construction of relativistic quantum fields [3] in the framework of white noise analysis, J. Math. Phys., 40 (1999), 5387-5405. T. Hida, H. H. Kuo, J. Potthoff and L. Streit, White Noise, An Infinite [4] Dimensional Analysis, Kluwer, 1992. [5] Z. Y. Huang, Quantum white noise, white noise approch to quantum stochastic calculus, Nagoya Math. J. 129 (1993), 23-42.

233

[6] Z. Y. Huang and S. L. Luo, Wick calculus of generalized operators and its applications to quantum stochastic calculus, InJinite Dimensional Analysis, Quantum Probabilty and Related Topics, 1 (1998), 455-466. [7] Z. Y. Huang and S. L. Luo, Quantum white noise and free fields, Infi-

nite Dimensional Analysis, Quantum Probability and Related Topics, 1 (1998), 69-82. [8] 2. Y. Huang and J. A. Yan, Introduction to Infinite Dimensional Stochastic Analysis, Scinece Press/Kluwer Academic Publishers, 2000. [9] S. L. Luo, Wick algebra of generalized operators involving quantum white noise, J . Operator Theory, 38 (1997), 367-368. [lo] N. Obata, An analytic characterization of symbols of operators on white noise functions, J. Math. SOC.Japan, 45 (1993), 421-445. [ll] N. Obata, White Noise Analysis and Fock Space, LN in Math., 1577, Springer-Verlag, 1994. 1121 E. P. Osipov, Quantum interaction : d4 :, the construction of quantum field defined as a bilinear form, J. Math. Phys., 41 (2000), 759-786. [13] S. M. Paneitz, J. Pederson, I. E. Segal and Z. Zhou, Singular operators on Boson fields as forms on spaces of entire functions on Hibert space, J. Funct. Anal., 100 (1991), 36-58. [14] J. Potthoff and L. Streit, Invariant states on random and quantum fields: &bounds and white noise analysis, J. Funct. Anal., 111 (1993), 295-331. [15] R. R3czka, The constructuion of solution of nonlinear relativistic wave equation in X : 44 : theory, J. Math. Phys., 16 (1975), 173-176. [16] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. I1 and 111, Academic Press, 1975. [17] B. Simon, The P(4)z Euclidean (Quantum) Field Theory, Princeton University Publisher, 1974. [18] W. Strauss, Nonlinear Wave Equations, CBMS Reg. Conf. ser. #73, AMS, Providence, R. I., 1989. [19] M. E. Taylor, Partial Differential Equations III Nolinear Equations, Springer-Verlag. New York Inc., 1996.

Generalized Horrnander theorem for non-local operators T. Komatsu and A. Takeuchi Department of Mathematics, Osaka City University email: [email protected]

Abstract

The smoothness of marginal distributions of solutions to

SDE's with jumps is investigated via the Malliavin calculus on the cbd-lhg space. A key lemma on the estimation of some functionals of semi-martingales is presented. Using the key lemma, the Hormander theorem on the hypo-ellipticity of parabolic differential operators is generalized to a theorem for the integrodifferential operators associated with the SDE's. This article is essentially a summary of [3]. But the last section where the main theorem is presented is an improvement of [3].

$1. Variation for SDE with jumps Let Ak = uk(x) . 8, ( 0 5 k 5 rn) be smooth vector fields on Rd,and let Be (0 E R") be operators defined by

Be$(x) = $ ( x

+ b(x,0))

(V E C W ( R d ) ) ,

where b(x,0) E C 1 ( R dx R" -+ Rd)with b(x,O)= 0 and b(x,0), &b(x,0) are smooth in x . Consider the integro-differential operator

where n ( d 0 ) = lei-"-* d0 (0 < a < 2). This operator is associated with a SDE driven by a LQvyprocess. Let W = D ( R + -+ Rm)3 w , Xt(w) = w(t) and W t , W be usual a-fields. Let (W,W ,P ) be the probability space characterized by properties that the counting measure J x ( d T d 0 ) = #{ t E 234

235

d r I 0 # A X , E d8 } gives a Poisson random measure with the intensity E p [ J x(drdB)] = ~ ( d 6 ) and d ~ the process

is a Brownian motion. Namely, the process (W,W ,W t ,P ; X t ) is a L6vy process such that, for any s < t and 8 E R”, ~

pe x [p [ G i e .

(x, x,)]I w , ] -

= exp[-(t - q 1 8 i 2 / 2

+ c&ia)].

We shall proceed the variational calculus on W . The functional derivative for functionals on W shall be defined not in the manner in [l]but in a similar manner to that in [ 2 ] . Let b’(z,6 ) denote the derivative d,b(z, 8) and assume that llb’(z,8)ll 5 c. < 1. Moreover assume that a k ( z ) and b(z,8) have following regularities

236

We shall define a fluctuated process X c = ( X : ) with a parameter by

E

E Rd

and consider the variation for functionals of XE. By the Girsanov theorem, the law Pc = P o (Xc)-' of the process X c and the probability P are mutually absolutely continuous on W t and

+

where 1'" = 1IC(xs,us) and kc = k~(x,,u,,O)= exp[-ahsf] det(1 O &(her)). We observe that the process xf := @ t ( X c ; x A )satisfies the equation

dxf

= ; i o < x ~dt)

+

c

ak(Xf)(d&

+l k f dt) +

1

b(xf,exp[h0(10) J X .

k

Set & ( X ) := (&)o x:. The process & ( X ) can be represented in the form

I"

&(x) = [%I-'

L ? ( x . , u s ) { c (usak(x,))(u,ak(x,))* ds

+

/

k (u.9b(xS,

e))(uSb(x81

e))*Jx}

([3] Lemma 2.1). We shall call & ( X ) "the Malliavin covariance".

92. Formulas of integration by parts

A function

Rd x R" is said to belong to the class C T > bif, for g p L v ( 0) x , = (aZ)@(O.&)vg(xl6 ) are bdd continuous

g(z,O) on

any ( p , v ) E Zd,", and satisfy SUP

(I

+/

/ s p L v ( x l O )a ( W 2

19PLv(X7O)l2

a(d6))


b. Let dF(O) be the class of all stochastic differentials & ( X ) of the

237

form

where fo(z), f k ( 2 ) E C">band g(x,6 ) E CF)b. We shall inductively define classes A process (yt(X))belongs to the class Y(")iff it satisfies a linear SDE

dYt(X) = (dFt(X))Yt(X)+ d+t(X) for certain dFt(X) E dF(O),d&(X) E dF("-'). Let dF(") be the class of all differentials dpt(X) of the form

where f k ( Z , x ' " ' ) (0 5 k) (resp. g ( x , q ' " ) , e ) ) are polynomial of random variables in the collection yt(n) :

Yp) =

(J { y:

; y(X) = ((&,y$,

. . . )*) E

Y(j)}

jln

multiplied by functions in C"2b (resp. C?)b). Set Mt(X) = (&)OM$. Then we observe that dMt(X) E d F ( l ) and dV,(X) E d F ( 2 ) . Lemma. ([3] Lemma 3.1, 3.2) For any integer n 2 1, if dyt(X) belongs to the class dF("-'), then pt(Xc) is differentiable in E and SUP EP[ SUP (IPs(XE)IP+ Ila,Ps(XE)llp)1 < 00 E

sst

f o r p > 1, and d((d/d&)o pt(Xe)) (1 5 j 5 d ) belong to the class dF("). Condition [V,]. &(X) i s non-singular and

Under the above condition, we shall define operators D l ( t ) , . . . , Dd(t) acting for smooth functionals F ( X ) on W by

238

and set D ( t )= ( D l ( t ) ,. . . , D d ( t ) ) . Let $(x) be a test function on Rd. Since E p [ $(xf) [&(Xc)-’]i M$ ] is independent of 6, we have

This can be generalized as “the formula of integration by parts” on W . Theorem 1. ([3] Theorem 1) Assume that Condition [&I is satisfied. For any p E Z:, the integrable random variable (D(t)”l)(X) is well-defined and

E P [ ( ( ~ z ) l L $ ) (=~-%[$(xt) t)l (w)”l)(x)l holds good f o r any test function 4(x) o n R d . From the Sobolev lemma, the above theorem implies that the law of the random variable xt has the smooth density w.r.t. the Lebesgue measure. Condition [Qt]. For any p > 1,

+ /A(1,

There exist positive constants

+

E,

b ( x , 8112 A 1 r ( d o ) } 2 c

6, c such that, f o r

x26.

We see that Condition [A] implies Condition [Qt]([3] Section 4), and expresses the non-degenerate property of the operator L. Roughly speaking, the hypoellipticity of the operator L holds under the Hormander condition on operators {Ao, A l , . . . ,Az,} (cf. [6], [7]).Though there are many cases

239

where Condition [A]is satisfied but the dimension of the Lie algebra generated by {Ao,A l , . . . , AZm} is less than d, Condition [A]is too much simply reduced from Condition [Qt]. The aim of this article is to present a generalized Hormander condition for the collection of operators Ak (0 k _< 2m) and Be (0 c R“).


1,

the inequality X2

I’

X2F(t)2A 1 d t + A-”

log MT(X,F )

+C

The proof of the key lemma is rather long. But the proof of the lemma for continuous semimartingales is quite short. Therefore the lemma enables us to give a new simple proof of the Hormander theorem for differential operators ([4]).Applying Key Lemma, Kunita [5]proved the existence of smooth densities of marginal distributions of canonical processes with jumps under a condition which resembles the Hormander condition.

240

For a vector field @ = $ ( x ) *

a,,let

The following lemma is proved immediately from Key Lemma. Lemma. For any 0 < u < 1/4, C E Rd with = 1 and @, there exist a positive random variable MT z MT[XO

where wj are independent flat Brownian motions (See [LW]). The law of zt(z)has the density p t ( z , y) with respect to 7r. Let TR(X) be the exit time of the ball B ( z ,R) of radius R and center 3: for the Carnot-Carathkodory distance. We get: Lemma 11: Under the previous assumptions, the measure p t , R : f + E[lTr(,) 0 and where d is the dimension of M . Proof: We consider k big enough in (3.14). The density of q f ( z , y ) is equal to pt(x,y) modulo exp[-C/t] for a very big C. We apply the techniques of [L3] (see [BAI] too) in order to show by using Malliavin Calculus that qf(x, y) has the asymptotic expansion (4.2).

0 Remark: We can apply this localization technic to the case where the two points are joined by a finite dimensional manifold of bicaracteristics (See [T.Wa]). Let us suppose from now that Xo(z)= 0 at the starting point (Hypothesis H(5)). Let N ( z ) be the grad of the Lie algebra spanned by the Xi,i > 0 and XO,XO alone excluded where we count 2 the weight of Xo. We get: Theorem IV.2: i)Let us suppose that Xo = 0 identically. Then there exists an asymptotic expansion (4.3) i=O

where t

+0

and where CO(Z)

> 0.

254

ii)Let us suppose only H(5). Then the asymptotic expansion (4.3) is still true, but we don’t know if cg(x) > 0. Otherwise, all the terms of the asymptotic expansion are 0. Proof: w e consider q t ( x , x ) the density of pk at x. We can apply the tools of Malliavin Calculus t o q t ( x , x ) . In particular, it checks (4.3) by the technics of [BA2], [TI and [L6]. Moreover, q;(x,x) differs from p t ( x , x ) by an exponentially small term. Therefore i). ii) comes from the same considerations by using the analoguous result of [BA.L] in the bounded case.

0 5. Non exponential decay without boundedness assumption We work now over Rd, and we suppose that we are in the situation ii) of Theorem IV.2 where the asymptotic expansion is trivial. We assume the following hypothesis (Hypothesis H(6)): there exist two real numbers strictly positive C and C’, an integer n and a real strictly positive constant K such that for all integers T , for all i, 0 I i I d

This hypothesis shows that in some sense, the vector fields belong to some generalized Gevrey class, such that the heat kernel when t -+ 0 belongs to a generalized Gevrey class, which allows us t o get an estimate of the decay of the heat kernel. This hypothesis is checked when the vector fields are polynomial near the starting point. We get the following theorem: Theorem V: There exists a real a0 > 1 such that for all t 5 1 p t ( 2 , x ) 5 Cexp[-llogtJao]

(54

Proof: We choose k enough small. g t ( x ,x) and p t ( z , x) differ by a term exponentially small. We can apply Malliavin Calculus to q t ( x ,x) and the technics of [F.L]in order to show that (5.3)

0 References [A]

Alexopoulos G.: Sub-Laplacians with drift on Lie groups of polynomial volume growth. Memoirs A.M.S 739 (2002).

255

Azencott R.: Grandes deviations et applications. Ecole de ProbabilitQsde Saint-Flour VIII. P. Hennequin edit. L.N.M. 774 (1980), 1-176. [BAI] Ben Arous G.: DQveloppement asymptotique du noyau de la chaleur hors du cut-locus. Ann. Sci. Eco. Norm. Sup. 21 (1988), 307-331. [BA2] Ben Arous G.: Dkveloppernent asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. Ann. Inst. Fourier. 39 (1989), 73-99. [BA.L] Ben Arous G. LQandre R.: DQcroissanceexponentielle du noyau de la chaleur sur la diagonale (11). Prob. Th. Rel. Fields. 90 (1991), 372-402. [C.K.S] Carlen E. Kusuoka S. Stroock D.W.: Upper bounds for symmetric transition functions. Ann. Inst. Henri. PoincarQ.23 (1987), 145-187. Davies E.B.: Heat kernel and spectral theory. Cambridge Tracts in Math. 92 (1990). Florchinger P. LQandre R. : DQcroissance non exponentielle du noyau du noyau de la chaleur. Prob. Th. Rel. Fields. 95 (1993), 237-267. Freidlin M.I. Wentzell A.D.: Random perturbations of dynamical systems. Springer. (1984). Hoermander L.: Hypoelliptic second order equation. Acta. Math. 119 (1967), 147-171. Ikeda N. Watanabe S.: Stochastic differential equations and diffusion processes. North-Holland (1981). Jones J.D.S. LQandre R.: A stochastic approach to the Dirac operator over the free loop space. In ”Loop spaces and groups of diffeomorphisms”. Proc. Steklov. Inst. 217 (1997), 253-282. Kusuoka S.: More recent theory of Malliavin Calculus. Sugaku 5 (1992), 155-173. Ldandre R.: Minoration en temps petit de la densite d’une diffusion dQgknkrQe.J. Funct. Ana. 74 (1987), 399-414. LQandreR.: Majoration en temps petit de la densite d’une diffusion dQgQnQrQe. Prob. Th. Rel. Fields. 74 (1987), 289-294. LQandre R.: Integration dans la fibre associQe a une diffusion dkgknkrQe.Prob .Th. Rel. Fields. 76 (1987), 341-358. Lkandre R.: Appliquations quantitatives et qualitatives du Calcul de Malliavin. In ”French-Japanese Seminar”. M. Mdtivier S. Watanabe edt. L.N.M. 1322 (1988), 109-133. English translation: [Az]

256

in ”Geometry of random motion”. R. Durrett M. Pinsky edit. Contemporary Math 75 (1988), 173-197. Lkandre R.: Strange behaviour of the heat kernel on the diagonal. In ”Stochastic processes, physics and geometry”. S. Albeverio edit. World Scientific (1990), 516-527. Lkandre R.: Dkveloppement asymptotique de la densitk d’une diffusion dkgknkrke. Forum Math. 4 (1992), 45-75. Lkandre R.: A simple proof of a large deviation theorem. In ”Stochastic analysis”. D. Nualart, M. Sanz-Sol4 edit. Prog. Prob. 32 Birkhauser (1993), 72-76. Lkandre R.: Brownian motion over a Kaehler manifold and elliptic genera of level N. In ”Stochastic analysis and applications in Physics”. R. Sknkor L. Streit edt. Nato Asie Series. 449 (1994), 193-219. L6andre It.: Uniform upper bounds for hypoelliptic kernels with drift. J. Math. Kyoto. Univ. 34 (1994), 263-271. Lkandre R.: Positivity theorem without compactness assumption. Preprint (2002). Meyer P.A.: Flot d’une kquation diffkrentielle stochastique. Skminaire de Probabilitks XV. J. Azkma M. Yor edit. L.N.M. 850 (1981), 100-117. Meyer P.A.: Le Calcul de Malliavin et un peu de pkdagogie. R.C.P. 25. Volume 34. Publ. Univ. Strasbourg (1984), 17-43. Nualart D.: The Malliavin Calculus and related topics. Springer (1995). Takanobu S.: Diagonal short time asymptotics of heat kernels for certain second order differential operator of Hoermander type. Public. RIMS. Kyoto Univ. 24 (1988), 169-203. [T.Wa] Takanobu S.Watanabe S.: Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integration. In ” Asymptotics problems in probability theory: Wiener functionals and asymptotics”. K.D. Elworthy N. Ikeda edit. Pitman. Res. Notes. Math. Series. 284 (1993), 194-241. [Wa] Watanabe S.: Stochastic analysis and its application. Sugaku 5 (1992), 51-71.

Precise estimations related t o large deviations Song Liang

Graduate School of Mathematics, Nagoya University, Nagoya, Japan

1. State of the problems In this paper, we are going to study some precise estimations of the Large Deviation Principles with respect t o the i.2.d. case and the diffusion case. Before stating the problem, let us look at an example, a simple model in statistical mechanics] to see why we need the precise estimations. Consider a system with n particles. We assume that for each particle] the set of the possible states is a compact metric space M , and when there is no interaction, the distribution of each particle is po, a probability measure on M . Also, we assume that the interaction between the particle in state x and the particle in state y is $V(x1y), where V is a continuous function on M x M . Then the probability of the system being in the state x = (XI,...,xn)is

where 2, is the normalizing constant. Many interested quantities in statistical mechanics such as the magnetic susceptibility and the correlation coefficients could be expressed as n

257

258

with some m E N and some function f . One wants to know (as n 4 m) if these quantities converge, and if converge, what are the limits. We can generalize this problem into the following. Let p be a probability measure on a separable real Banach space ( B ,11 . II), satisfying the condition that there exists a constant C > 0 such that JBexp(Cllzl12)p(dz) < 00. Also, let @ be a continuous function of B , which can be dominated by a first order polynomial of ll~cllfrom above. Let 2,

EP@m[exp(n@(~Sn)], where S,

continuous function F : B

= Cy="=,i. For

any bounded

R, one wants to know the precise asymptotic behavior of Z;lEP@m [F(n-'S,) exp(n@(n-'S,))]. The problems about -+

the magnetic susceptibility and so on in the example above can be resolved into this problem if we let B = C ( M ) * and let p be the distribution of

6, under po(dz), where 6. denotes the delta measure. ( B is not separable unless M is countable, but the same argument is still valid to give us the same result). By Large Deviation Principle (c.f.,Donsker-Varadhan [S]), we know that

- h ( z ) }(denote it by A), where h is the limn-m log 2, = supZEB{@(z) entropy function given by

B' being the dual space of B , and M ( . ) given by M ( $ ) = JB e @ ( l ) p ( d z ) , $ E B'. The same could be said about the quantity

EP@-[F(n-' S,) e~p(n@(n-~S,))] . However, this is not enough to give a good answer of our former questions, and we need some more precise estimation of 2, as n 400. Similarly, we can consider the same problem for the diffusion process case. Let Td = Rd/Zd denote the d-dimension torus. Let

be

{P,}zETd

the family of probability measures on the path space R = C([0,00);Td), with infinitesimal generator LO = + A

+ bo . V, where bo

: Td

Rd is smooth. Let X , ( w ) = w ( t ) , t 2 0, and for any T > 0, let LT = $ So 6x,dt, a probability measure on Td. Let M(Td) = C(Td)*,the dual space of C(Td),which is also the set of all signed measures on Td with finite total 4

T

259

variation. Let Q, : M(Td)-+ R be a bounded and continuous function, and for any

2,y E

Td, let

We are interested in the behavior of

22'

as T

-+

ca.

We have by Donsker-Varadhan [9] that 1 log 2;' --+

T for every x , y E Td,where X = sup{Q,(v) - I(v);v E p(Td)}and I is given by

As before, we want t o know some more precise behavior of 2;" as T

4

03.

The rest of this paper is organized as following. We discuss about the

i.i.d. case in Sections 2 and 3, o(1) ordered behavior in Section 2, and the asymptotic expansion in Section 3. In Section 4, we are going to deal with the case of the diffusion process on torus, and an o(1) ordered estimation is given. There are many other related problems, such as the asymptotic expansion for the diffusion case (on torus), the same problems for the diffusion processes on Euclidean spaces, etc, which are also being studied, but will not be included in this paper. Also, we only discuss about the non-degenerate case in this paper. For the degenerate case, c.f. Liang [16], Bolthausen [5], Chiyonobu [7], etc, which gave some partial answers for the i.2.d. case, saying that 2, could be expressed as a integral on a finite dimensional manifold, which includes all of the degeneracy. The problems for the degenerate case are also being considered, and under construction now.

2. o(1) Order estimations for the i . i . d . case In this and the next sections, we are going to deal with the sums of the

i.i.d. Banach space-valued random variables. Let us state the setting of the problem, again, more precise this time, for the sake of clearness.

260

Let B be a real separable Banach space with norm

11. 1 1,

p be a probability

measure on B . We assume that the smallest closed affined space that contains suppp is B . Moreover we assume the following

(Al) There exists a constant Co > 0 such that exp(Co11x112)p(dZ)< a. Note that this is stronger than the condition any C

"

sBeCIIxIIp(dx)
0 ", which is one of the typical assumption for LDP. ( A l ) is used

essentially in the proof of Kusuoka-Liang [la], and we do not know whether (A1) can be weaken. Let @ : B

4

R be a three times continuously FrQchet

differentiable function satisfying the following: (A2) There exist constants C1, C2

> 0 such that

a(.) 5 C1 + C2llxll,

for any x

c B.

X, and S,, n E N, be the random variables defined by X n ( g ) = x, and s,(:) = C z = , X k for any:= (x1,x2,x3,...) E BN. Let

We are interested in the behavior of

as n + 00, where E X stands for the expectation with respect t o the measure A, and p@mis the infinite product measure of p on BN.

As stated in Section 1, we have by Large Deviation Principle (c.f., Donsker-Var adhan [8])that 1

lim -log 2, n

n-m

=

sup{@(x)- h ( x ) } , xEB

where h is the entropy function of p:

B* is the dual Banach space of B and M ( $ ) =

sBe + ( " ) p ( d z )for any 4

E

B*.

We will give a more precise estimation of Z,, up t o order o(l), in this section. Bolthausen [4] studied the same question under the "Central Limit Theorem Assumption". (c.f., Theorem 2.1 below).

261

It is not difficult to see that there exists at least one x* E B that attains the maximum of @-h, and the set of the elements that attain the maximum is compact. (c.f. Bolthausen [4] for the proof). We assume the following. (A3) There is a unique x* E B with

a(%*)- h(x*) = s ~ p , ~ B { @ ' (-x )

h ( x )1. We will use z* exclusively for this point. Let v be the probability measure on B given by

v(dx) = exp(D@(x*)(z))Pu(dx) M(D@(x*))

sB

It is shown by Bolthausen [4] that rcv(dx) = x*. Let vo be the O-centered 1 v , i.e. vo = v8, , where 8, : B -+ B is defined by Q,(x) = x - a , x E B . Let r(p,1cI) = (~(x)$(x)vo(dx)be the covariance of 'p and ?1, for any 'p,+ E B*. r is an inner product on B*. Let H = (Dr)*, where Pr means the completion of B* with respect t o r. Then it is not difficult that H can be regarded as a dense subset of B. It has been shown by Bolthausen 141 that the following holds:

sB

where

~ ( 4= ) sB$(x)xvo(dx),q5 E B*.

eigenvalues of the operator D2@(x*)I Furthermore we assume the following

From this, we see that all of the

HxH

are smaller than or equal to 1.

(A4) All of the eigenvalues of D 2 @ ( x * ) / Bolthausen [4] showed the following.

HxH

Theorem 2.1 (Bolthausen). Assume ( A l ) the following.

are smaller than 1.

-

(A4). Moreover, assume

(I?) vo satisfies a central limit theorem, i e . , v, defined v ; " ( f i A ) converges weakly to a Gaussian measure y on B . Then

by vn(A) =

262

Kusuoka-Liang [12] considered the same problem, they showed that if we assume the following condition with respect t o the third F'r6chet derivative of

a, the similar result

still holds without the Central Limit Theorem

Assumption (B). (c.f. Theorem 2.2 below). (A5) There exist a constant 6

K :B x B

-+

> 0 and a continuous bilinear function

R such that lD3@(4(Y,Y,Y)IF IIYIIK(Y,Y)

for any y E B and any x E B with IJx- x*IJ< 6. The following is by Kusuoka-Liang [12].

-

(AS)

~ 2 @ ( x * ) ) - '(= ~

2 ) .

Theorem 2.2 (Kusuoka-Liang). Under the assumptions ( A l ) above, we have

= exp

(i

1

~ a ( x * ) (x)vo(dx) x, . deb (IH

-

Remark 2.3. If we assume that vo satisfies central limit theorem as in Bolthausen [4],then ( H ,B , y) becomes an abstract Wiener Space, so by Kuo

( * I '@ ) HxH In this situation, the intergration K1 appeared in

[15] (Page 83, Theorem 4.6 (Goodman)), we can get that is a nuclear function.

D

Theorem 2.1 is nothing but the constant K2 in Theorem 2.2. However,

I

when the operator D2@(z*)

HxH

is not nuclear, (but it is not difficult that

under our present setting, it is always a Hilbert-Schmidt function), K1 is not defined, while

K2

is still well-defined.

As mentioned above, the central limit theorem assumption is actually a quite strong assumption as we are dealing with infinite dimensional space. Theorem 2.2 claims that even without the assumption that vo satisfies central limit theorem, the result still holds under the assumption (A5).

3. Asymptotic expansions for the i . i . d . case We use the same setting and same notations in this section as in Section 2. As claimed in Remark 2.3, we have that D2@(x*)I Schmidt operator. Let

ak E

R, e k

HxH

E

H * ,k

E

is a Hilbert-

N, be the eigenvalues and the

263

corresponding eigenfunctions of it. It is not difficult from the continuity of D 2 @ ( x * ): B x B + R that by extension, we can always assume that ek E

B* for any k E N with

ak

# 0. So we have that 00

D2@(z*)(z,Y) =

(3.1)

akek(z)ek(Y) k=l

holds for all z,y E H . Also, (3.1) holds for all z,y E

B if

xEl lakl < 00,

ie., if D Z Q ( ~ * ) ~ is a nuclear operator. HxH

We assume the following. (A6) There exists a bilinear, symmetric, bounded function

: B x B -+

R, and a monotone non-increasing sequence of positive numbers 6 ~N , E N , that converges t o 0 as N -+ 00 such that for any N E N, k >N,ak >0

cEl laklek 8 ek is welldefined and gives a continuous operator on B x B. Therefore, cp=llakl < R e m a r k 3.1. The assumption (A6) implies that

00,

ie., D 2 @ ( z * ) /

HxH

is a nuclear operator. Hence (3.1) holds for

VO-

B. Moreover, the constant K2 in Theorem 2.2 is equal to det ( I H - D2@(z*))-" (where the "det" stands for the Fredholm determias.

z,y E

nant). (A5') @ is 5-times continuously Frkchet differentiable, moreover, there exists a constant b > 0 and a bilinear, symmetric, bounded function K5 :

BxB

+R

such that

for any z E B with

< 6 and any y

112 - 2 * l l ~

E

B.

We remark that although this can not imply the condition (As), by checking the proof in Kusuoka-Liang [12] carefully, one easily sees that this is enough for Theorem 2.2. Albeverio-Liang [l] proved the following, which provides a precise expression for the coefficient of the term n-' exp(-n(@(z*) - h(rc*)))Z,.

in the expansion of U,

=

264

Theorem 3.2. Under the assumptions (Al)

-

(A4), (A57, (AS), we have

that

(

lim n exp(-n(@(x*) - h(x*)))Zn- det(lH - 92)-’/’) = C2(1c*),

n-cc

where Cz(x*) is a constant determined by Da@(x*), i = 2 , 3 , 4 .

See

Albeverio-Liang (11f o r the explicit expression of C,(x*). Remark 3.3. As remarked before Theorem 3.2, Theorem 3.2 gives the coefficient of the term n-l in the expansion of Un = exp(-n(@(Ic*) - h(x*)))Zn in powers of n-’/’-term

(i) Note that Theorem 3.2 also gives us that there is no

in the expansion. This comes essentially from the symmetricity

of Gaussian measure. (c.f., [l,Theorem 3.14, Example 3.151). By using the same method, one can get the explicit expression of the coefficient CN(X*) of the term n-N/2 for any N 2 2 in the expansion of

U,,

under natural assumptions about the smoothness of @ and some assumption corresponding t o (A5’). Let us explain the idea roughly in the following. Actually, by using the Taylor expansion, one gets the exponent of some finite sum of the terms n-4+lDk@(x*), k 2 2. The terms for k

23

are

+ 1 < 0 as long as k 2 3. The most difficult part

relatively easy, since

-$

is the one with k

2, but this is already done in [l].Albeverio-Liang [l]

=

did not write this explicit expansion, because of its complication. But Albeverio-Liang [I]did give the expansion, to all orders in n-N/2, of the term

for

E

>0

small enough, with controls on remainders. (See Theorem 3.14

there). Remark 3.4. In the sense explained in Remark 3.3, we have got an analogue of the Edgeworth expansion for the functional exp(:92), with

9 2

a bilinear,

symmetric and bounded function on B x B that satisfies the conditions (A4) and (A6), of the normalized sum variables Sn/&. The Edgeworth expansion with respect t o the distribution function of the functions of Sn/*

has

been done by many authors (see, e.g., Gnedenko-Kolmogorov [lo],Hall [ll],

265

Bentkus-Gotze [3], and references therein), but all of these give only estimations which are uniform with respect to the variable of the distribution function, and are not usable in the case of our problem (because of the lack of integrability of the function exp with respect t o the Lebesgue measure). In fact, in the expression

- P(Y I

@2(Y,Y)I n&))dY

+ n e - n ~ / 2 ~ @ ” ( 1S~ 2-)I Sn ( 2 , I n&) -nE

-

[ :

-

&J;;

exp(-Qz(Y,Y)), IQz(Y,Y)I

2

n e - n a / 2 ~ ( ~ * 2 ( 5~ n&) ,~)~

1

,

all the terms in the right hand side except the first one decay exponentially as n

03.

n ~-)P ( y

However, a uniform estimation of n(P”Fw(y 5

5 QJ2(Y,Y)5

TIE))

QJz(3, 5)5

with respect t o y E [ - n ~ , nis~not ] enough

for give any estimation of the first term, hence not enough to derive our asymptotic expansion. 4. o(1) Order estimations for diffusion processes on toruses

In this section, we are going t o discuss about the case with respect to the diffusion processes on torus. Precisely, we consider the torus Td = Rd/Zd, which is a compact manifold. The tangent space T(Td) can be identified with Rd. Let B(Td)be the set of all Bore1 sets in Td. Let M(Td) be the dual space of C(Td). M(Td)is the set of all signed measures on Td with finite total variation, and denote the norm derived by it, the total variation norm, by

11.11.

We also think of the weak*-topology in

M(Td). Let p(Td)and Mo(Td)be the set of all probability measures on Td and the set of all signed measures on Td with total measure 0, respectively. Let dist(., .) denote the Prohorov metric on p(Td). Note that the topology

266

induced by the Prohorov metric and the weak*-topology coincide. The path space R

C([O,oo),Td) is the set of continuous functions w : [0,oo) + T d . Let Xt(w)= w ( t ) ,t 2 0, let Ft = a { w ( s ) ;s 5 t } , and let F = VtFt. Let LO = ;A + bo ' V, where bo : Td 4 Rd is a C" function. Let {Pz}2ETd be the family of probability measures on 0 of the martingale problem LO,ie., for any z E Td and any f E CDO(Td;R), =

(2) P,(wo = z) = 1. Denote the corresponding semigroup of linear operators on C(Td) by

{Pt}t?o. { P x } z E T d has a unique invariant probability measure p , which is absolutely continuous with respect to the Riemann volume on Td, and

& is a strictly positive smooth function.

For any T

> 0, the distribution

law of {XT-t(w)}o 0 : z E XK},

( ( A ( (= K sup ( ( ~ ( I K = inf{X > 0 : A C XK}, xEA

K O

= {z* E

X* : SUP I(z*,x)l5 I}, xEK

J1z*IIp= inf{X

> 0 : z* E XK"} = sup I(z*,z)l, xEK

Note that K is bounded and

1 1 2

- yllx 5 l l K l l ~ d ~ ( z , yThe ) . set KO c X*

is called a polar of K and is a convex symmetric set. Let S* be the closed unit ball {z* E

X* : ( ( z * ( (_

A&,(K),there exists a constant Cz,s,asuch that for all w E C T ( A ' ( M ) ) ,

(3) For all p 2 2 and a > A&(K), there exists a constant Cp,a such that f o r all f E C F ( M ) ,

Remark 5.1. By 139, 431, the function p

4

Ap(A/2

+ K ) is decreasing on

[a,+a]. Let y ( K ) := sup Ile-t(W+K)

I _,_

.

O: v ing X,(A/2

E

T,M,

+ V2q5(z),

llz~ll= l}, V z E M .

+ K ) by X,(L/2 + K+):= - t-cc lim +log

Ile-t(L/2+K4)

Replacwe

can reformulate Theorem 5.1 and Theorem 5.2 for the Littlewood-PaleyStein functions

gZ,,,L(f),

gl,a,L(w)

and for the Riesz transform R,(L) =

V ( a + L)-?

6. Manifolds satisfying the Sobolev inequality By Theorem 5.2, the finiteness of X,(A/2

+ K ) (the Lm-bottom of spec-

+ K ) yields the LP-boundedness V ( a + A) for all p 2 2 and a >

trum of the Schrodinger operator A/2 of the Riesz transform &(A) max{-X,(A/2

+ K),O}.

=

In the literature, see e.g. [8], the condition

+ K ) > -m is called the gaugeability of the Schrodinger operator A/2 + K . In general, it is not easy to verify that the Ricci potential X,(A/2

K satisfies the gaugeability condition if we do not assume some further geometric conditions on M . In this section, we give a gaugeability result of Schrodinger operators on complete Riemannian manifolds satisfying the Sobolev inequality. Our result leads us to find a new class of complete Riemannian manifolds with unbounded Ricci curvature on which the Riesz transform Ra(A)is bounded in LP for all p

2 2 and a 2 0.

Recall that, see e.g. [36], for any complete Riemannian manifold M with positive injectivity radius, there exist two constants A and B such that the

302

following Sobolev inequality ( S A B )holds: for any f E Cr(M),

Ilf

5 Allof 1 ;

1 1 n--2 2 L

+ Bllf 11%

n = dimM 2 3.

(SAB)

For manifolds with certain curvature assumptions and sufficient large volume growth at infinity, B can be set equal to zero. By [44, 131, the Sobolev inequality ( S A B )implies that the heat kernel of the Laplace-Beltrami operator A on M satisfies p t ( z ,y ) 5 c t - 4 ,

v 0 < t 5 1.

Moreover, the Riesz-Thorin interpolation inequality yields for all q

2 1,

Modifying an argument as used in Davies and Simon [14], we can prove the following Lemma 6.1 (( [ 2 5 ] ) ) . Let M be a complete Riemannian manifold on which the Sobolev inequality ( S A B )holds. Suppose that K+ E L"

and

( K - $ ( K ) ) - E L4+" for some E > (2 - :)+, where XB(K) := max(0, X2(A/2 + K ) } . Then X,(A/2

+ K ) is independent of p E [2, m].

By Lemma 6.1 and using an earlier result due to Rosenberg-Yang [36] on the positivity of Xz(A/2

+ K ) , we can prove the following

Theorem 6.1. Let M be a complete Riemannian manifold satisfying the

, = d i m M 2 3. Sobolev inequality ( S A B ) n (1) For n 2 5 , suppose that K+ E L", K - E L4+€ f o r some

there exists a constant

LY

> 0 such that

E

> 0, and

303

T h e n X,(A/2

+ K ) > 0 , and Ro(A) = VA-'I2 is bounded in LP f o r all

P>2. (2) Suppose that (6.1) holds (for n = 3,4) or there exist some constants

q > n, a > 0 and C(n,q ) > 0 such that

Suppose further that K+ E L",

K-

E

Ls+' f o r some

E

> 0 and

~ ~ K - ~ 2, where

[

~ ~ K - ~:= ~ v S, U w P V O{ X~ E M : K - ( x ) 2 (}(" c-0

Then X,(A/2

1

. l/"

+ K ) 2 0 , and & ( A ) = V(u + A)-ll2 is bounded in LP for

a l l p 2 2 anda>O. Remark 6.2. By [26],if we assume Vol(M) = +a, then X,(A/2+K)

>0

implies that the Ricci curvature on M cannot be uniformly bounded from below. Thus, Theorem 6.1 provides us with a class of complete non-compact Riemannian manifolds with infinite volume on which the Ricci curvature is not uniformly bounded from below while the Riesz transform & ( A ) =

V ( a + A)-1/2is bounded in LP for all p 2 2 and a 2 0. Remark 6.3. In [9], Coulhon and Duong have given a counter-example of complete non-compact Riemannian manifolds with small portion of negative Ricci curvature and with infinite volume on which the Riesz transform

& ( A ) = VAP1I2is not bounded in LP for sufficient large p > 2. In their case, the Sobolev inequality ( S A B )holds with B t o see that for every a

= 0.

However, it is easy

> 0, ll(K - a ) - l l ~= +m since K is identically zero

out off a compact subset.

Acknowledgement. The author is grateful to Professors A. Ancona,

D. Bakry, T. Coulhon, D. Elworthy, E. Hsu, M. Ledoux, X. -M. Li, N.

304

LohouB, T. Lyons, P. Malliavin and Feng-Yu Wang for helpful discussions and comments. He also thanks anonymous referee for his careful reading and for helpful suggestion to improve the paper.

Final Remark in May 2004. This paper is an announcement of the results obtained by the author in [25] during April 2001- April 2002. These results have been further improved in the author’s new submitted preprint ”Riesz transforms associated with diffusion operators on complete Riemannian manifolds”. Due t o this reason, [25] will not be submitted for publication. We refer the reader t o the new preprint for the details of the proofs of the results in this paper.

References [l]D. Bakry, Transformations de Riesz pour les semi-groupes symm-

Qtriques, S6m. Prob. XIX, Lect. Notes in Maths., 1123(1985), 130-175.

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Anal., 63 (1985), 240-260.

The ground state in Nelson’s model with or without infrared cutoff J6zsef LBrinczi

Zentrum Mathematik, Technische Universitat Munchen

85747 Garching bei Miinchen, Germany [email protected]

Abstract Based on recent work, we discuss how the problem of infrared divergence in Nelson’s massless scalar field model can be tackled. We translate the problem originally formulated in terms of operators into a problem of stochastics by associating well specified processes to the particle and field operators. The ground state will be put in direct relationship with data given by Gibbs measures relative to these processes.

When

interpreted back, it turns out that in three dimensions the Nelson Hamiltonian has no ground state in Fock space and thus is not unitary equivalent with the Hamiltonian obtained from Euclidean quantization. In contrast, for dimensions higher than three the Nelson Hamiltonian does have a unique ground state in Fock space and the two Hamiltonians are unitary equivalent. We show how another representation for the 3D case can be constructed which eliminates infrared problems altogether. Keywords. Nelson’s scalar field model, infrared divergence, ground state, Gibbs measure 309

310

1. I n t r o d u c t i o n As well known from standard texts, when the momentum of a charged particle is suddenly changed, i t releases a certain amount of energy carried by photons. It may happen that the total number of these soft photons diverges while their total energy stays finite. These are the basic symptoms of the so called “infrared catastrophes” which occur commonly in such situations. Here we present the case of a quantum particle coupled to a Bose field and show how the problem of IR divergences can be solved. The problem will appear as non-existence of a ground state for this system in the product (standing for space of Fock space (standing for the field) and L2(Rd,dz) the particle). The direct way of studying ground states is to look at the eigenvalue equation. Another one, adopted here, is t o look at the semigroup defined by the exponential of the Hamiltonian. Then, by a Trotter-type argument, a relationship can be established between scalar products (ground state expectations) on the product space just mentioned, and expectation values with respect to a certain probability measure on an associated function space. As it turns out, the latter approach is particularly useful in this case. The particle-field interacting system is described by the Hamiltonian

This is a generalized Nelson Hamiltonian whose particular cases depend on the choice of w as will be done below. Here we use standard (though somewhat difficult) notation from quantum field theory which would require a detailed mathematical presentation, but soon will turn the problem into a completely different language, thus we only recall their meaning. In the first

311

term we havc thc Schrodinger operator H p = -(1/2)A

+ V on L2(IWd,dq)

for the quantum particle. V is assumed to be a confining potential growing a t infinity, making Hp have a purely discrete spectrum. More precisely, we allow two classes of them:

(Vl) V is bounded from below, continuous, with asymptotics

for large 141, with some constant C > 0 and exponent a

> 1.

(V2) V is of Kato-class bounded from below as

with some constant C

> 0 and exponent a > 0.

Not everything we want to prove can we do for (V2), therefore we will sometimes use the more restricted class (Vl). The second term in (1.1) is the energy of the Bose field with dispersion relation w(lc) 2 0. For a shorthand we put Hf

= Jwd

w ( l c ) a * ( k ) a ( k ) d k Here . a * ( k ) , a ( k )are the Bose

creation and annihilation operators acting on Fsym, the symmetric Fock space over L2(Rd,dlc). The last term describes a shift invariant interaction between the particle and the field, and it is significant that it is linear in the field operators. p stands for the charge distribution making the coupling between particle and field smooth enough. We assume it to be a smooth function, spherically symmetric, and fast decaying. Hats will denote throughout Fourier transforms. The normalization JWdp(z)dz = e 2 0 calibrates the strength of the interaction.

HgNthus acts on L2(Rd,dq) 8 Fsym. Under

the assumption that

312

H g is~ bounded from below, and the interaction is infinitesimally Hfbounded. Therefore H g is~ a self-adjoint operator with domain

D(H,8

1)n D(1 CZJHf).

The charge distribution brings in an ultraviolet cut-off. Ideally we would

like to understand the ~(x)--+

e&(z) (or,

equivalently, G ( k ) 4 ( 2 7 ~ ) - ~ / ' e )

cutoff-free case. However, that requires a very different kind of analysis [3] and here we will assume such a cut-off and only study the behaviour at the other end of the spectrum. The latter is of interest for massless bosons, i.e.

w ( k ) = lkl. We refer to HgNwith this dispersion relation as the massless

Nelson model and denote the corresponding Hamiltonian by H,. For d = 1 or 2 it is not bounded from below, thus we require d 2 3. The infrared cut-off is introduced by the requirement that

As shown in [7], under this cut-off the general Nelson Hamiltonian has such a unique, strictly positive ground state, i.e. a \k E L2(Rd,dq)8 FSym = EoQ, where Eo = inf Spec(H,,). By now we have a detailed that HgN@

understanding of the properties of the ground state given that (1.5) holds, see [ 2 ] . What we discuss here is whether there can be found a ground state in the same space or elsewhere when no infrared cutoff is imposed. We reached our goals in [2] by using a stochastic rather than analytic technique. In this so called Euclidean picture there are no operators as above but stochastic processes on suitable function spaces with trajectories

t

M

qt of the particle, resp. t

H

&(x), x

E Rd, of the field as primary

input. Formally, the Euclidean action for the particle is

313

and

for the field. From (1.1)the particle-field interaction restricted to the time interval [-T, T ] Sint,T({qt,Et})

=

sT/ -T

Wd

T

e(z - qt)S(t,z)dzdt = S_,(E' * e)(qt)dt

(1.8)

follows (* denoting convolution). Thus, heuristically, the full Euclidean path measure is

As might be already guessed, this path measure should be able to inform us about relevant properties of the original system. First we turn to giving proper definitions to the so emerging processes.

2. Basic processes for particle and field In either of the cases (Vl) or (V2) above Hp has discrete spectrum and a unique, strictly positive ground state i.e.

$0

lying at its bottom Ep,

HP$o = Ep$o. The ground state transform is a unitary map

L 2 ( R d$$dq) ,

-+

L2(Rd,dq), Hp H $CIHp$o. This generates the Rd-valued

Ito-process {qt : t E

R}

(also called P(q5)I-process on path space C(IR,Rd))

defined by the stochastic differential equation

where dBt is Brownian motion. The path measure of this process will be denoted by

N o and

its stationary measure by dNo = $ i ( q ) d q . This path

measure can be understood as a Gibbs measure arising from the modification of two-sided Wiener measure by densities dependent on the potential

V (note that in the formulas above V is invisible but is in fact there through

314

40). The paths Q = {qt

: t E IR} are No-almost surely continuous, and the

process is Markovian. A typical path grows like

with suitable C1 > 0, C2 = C2(Q), and

a!

being the exponent in ( V l ) , resp.

(V2); for more details see [l]and references listed there. Next, consider the Schwartz space S’(Rd) of tempered distributions and the space S(IRd)of real-valued test functions. The process associated with the free field is the S‘(Rd)-valued Ornstein-Uhlenbeck process {& : t E with path measure

for all such

f

E

9 , the Gaussian

R}

measure of mean zero and covariance

S(Rd) that have sIf^(k)12/w(k)dk
0. We denote the t

= 0 distributions for

p and Po by P and Po, respectively. We are interested whether the sequence of measures PT has a limit as

T

-+

00.

The following notion of convergence will be used. Let M be a

metric space, and C(R, M ) the space of continuous paths { q t } with values in M . For any interval IT = [-T,T] c R let M T of the Bore1 a-field

MT

= cT(Xt,t E

c generated

c C be

a sub-a field

by the evaluations { X t : t E I T } , i.e.

IT). We say that a sequence of probability measures {pn}

on C(R,M) converges locally weakly to the probability measure p if for any 0 5 T

< rn the restrictions pLnlMT converge weakly to the measure

p l ~ We ~ have . then the following result.

Theorem 2.1. Let d 2 3, the coupling constant be 0 < e 5 e * , with suficiently small e* > 0, and V satisfy (Vl). Then there exists the local weak limit pT -+P as T -+00. p is a probability measure of a stationary reversible Markov process o n the path space C(R,Rd x B D ) . Moreover, P can be obtained as the weak limit P = limT,,

PT.

316

For a detailed proof see [2]. For any path Q = {qt : t E R} E C(R,Rd) the conditional measure

is a probability measure on C(R, BD),and the measure PT on C(R,R d x BD) can be written as the mixture PT(S

Here A

x A) =

c C(R,BD) and

S,

P T & ( A ) ~ N T ( Q ) , VT

> 0.

(2.10)

S c C(R,Rd), and NT is the particle path-

marginal of PT. The crucial idea is that since the process

& is Gaussian

and the coupling is linear in it, this marginal can be computed explicitly and is given by

with the same pair potential as in (2.8). ~l$ is a Gibbs measure for the bounded interval [-T, TI, with reference measure N o ,i.e. the path measure describing the free particle.

P?

is itself a Gaussian measure, thus the only

difficulty in understanding the joint path measure lies in the Gibbs measure

NT. In order to complete the argument a good control of the NT + N limit is thus needed. This problem with a scope going beyond Nelson's model has been investigated in [4]by a cluster expansion technique and some of the results relevant in our context are quoted next.

Theorem 2.2. For d 2 3, suficiently weak coupling e > 0 , and potentials

satisfying ( V l ) the following properties hold: ( 1 ) There exists the local weak limit NT,

T,

i00

+

N f o r all positive sequences

(independent of their choice).

(2) The measure h/ is invariant with respect to time shifts and time

reflections.

317

(3) The t = 0 distributions N T , resp. N, of the measures

NT, resp.

N , are mutually absolutely continuous with respect to NO, i.e. there exists a constant c > 0 such that of

(2.12)

uniformly in T and 4 0 . Moreover, (2.13)

(4)

For No-almost all q the conditional distribution NT( . lqo = q )

converges weakly locally to n/( . Iqo = 4 ) . (5) N - a l m o s t all paths Q = { q t } E C ( R ,Rd) have the property that for

all t E R

where

C1

> 0 i s a constant, Cz i s a function of the path Q , and a

is the growth exponent of the potential appearing in (Vl). (6) For any bounded functions F1, F2 o n Rd the following estimate

holds o n their covariance:

(2.15)

with some y > 0 and a suitable prefactor C > 0 .

3. The infrared divergent case: d = 3 Aker having introduced the path measures for the interacting system, we turn now t o compare the Fock quantization and the Euclidean quantization of the interacting system. The first one is constructed through the Feynman-Kac formula,

318

which defines the semigroup exp(-tH) on

3-1'

=

L2(Rdx B ~ , d p ' ) . Its

generator, the Hamiltonian H , is unitary equivalent with H N . For the Euclidean quantization we start with the semigroup Tt associated with the time reversible Markov process { q t , &} distributed by P as defined through

on the Hilbert space 3-1 = L2(Rdx l ? ~ , d P ) .Tt is a symmetric contracting semigroup, hence there exists a self-adjoint semibounded operator He,, generating it, i.e. Tt = exp(-tH,,,),

which by definition will be viewed as

the Hamiltonian of the system obtained in Euclidean quantization. Note that the constant function 1 is a ground state of He,,, i.e. an eigenfunction lying at the bottom of its spectrum. Since Tt can be shown to be positivity improving, this ground state is actually unique in 3-1. In this and the following section we investigate whether the Hamiltonians

H - Eo and He,, are unitary equivalent. We use throughout the following Definition 3.1. The Hamiltonian H is called infrared divergent if it has no ground state in 3-1'. Let {PA} be the family of spectral projections for the self-adjoint operator H bounded from below, and denote dal(X) = d(1, P x l ) % o , the spectral measure for 1 E 3-1'.

Then

EO 5 E; := inf suppdal(X).

(3.3)

Eo is the bottom of the spectrum of H . We also define the approximate ground state

Theorem 3.1. Suppose V satisfies condition (V2). Then H is infrared divergent if and only ijlimT-+m(l,Q

T ) ~= o 0.

319

This theorem results directly from

Lemma 3.2. W e have the following two cases:

(i)

If lim supT--tm(1,~ T ) % H>O0, then 01 ({EA})> 0 and the limit lim 9~ := 9.

(3.5)

T-+W

exists. Moreover, 9 > 0, EL = Eo, and 9 is the unique ground state of H at eigenvalue Eo . (ii) If limsupT+m(l, Q\I~T)%O = 0, then al({EA})= 0 . Moreover, EA

=

EO

and H has no ground state in 'H0. For details of proof see [ 5 ] .

A virtue of the stochastic approach is that IR divergence can be directly related t o the properties of the path measure:

Lemma 3.3. For d

=

3, ( V l ) potentials, and weak enough couplings

0 < e 0 and IEp[F] = 0.

Here, information on the typical

paths of the Gibbs measure becomes relevant, that is why we take (Vl) to use the result (2.14) in Theorem 2.2 obtained by cluster expansion. The second assertion of the lemma comes about by Schwarz's inequality applied to this particular F to get (QT,1)%0 =

J' 9 T d P o = J' \kTF1/2F-1/2dP0 5

(1

9?j-FdPo) '/'

(1

F-'dP0)

(3.6)

1/2

.

As by the first assertion of the lemma the first integral at the right hand side above converges to zero as T

-+

00,

and since the second integral is

320

bounded, the claim follows.

A combination of Theorem 3.1 and Lemma 3.3 leads t o the main theorem of this section:

Theorem 3.4. For d

=

3, (V1)-potentials and 0 < e 5 e* the Hamil-

tonian H has n o ground state in 'Ho, and hence H N has n o ground state

in L 2 ( R d , d q ) 8 3ssym. In particular, H - Eo and He,, are not unitary equavalent. The restriction to weak coupling is of technical nature only. In order t o define 'H and He,, we first had to construct the Gibbs measure

N for the interacting system which required the coupling to be sufficiently weak. In addition, we used that the fluctuations of qt are logarithmic. If we only want to show that H has no ground state in 'Ha, then here is an alternative proof avoiding all restriction on the coupling constant, and moreover valid for the larger class (V2) of potentials.

Theorem 3.5. Suppose

e20

and V is of class (VZ). T h e n f o r d = 3 the

Hamiltonian H has n o ground state in 'Ha. Proof: For any finite T > 0 the measure PT is absolutely continuous with respect to P o , and

(3.7) where

2,- is defined by changing

2,-

=

ST for f , in the exponent,

and obviously

2;. By applying the Feynman-Kac formula we obtain

321

in the norm of L2(Rd x B ~ , d p ’ ) .Thus

(3.11) Now by (3.7), (3.8) and (3.9) we have that

(3.12) Furthermore ~ p[ ~ o

where

N:

01=

$(q,

T

FN:

IENO

(3.13)

[IEp [e- So ( ~ t * e ) ( q t ) d t140 ] =q]]

is the measure for paths in the forward direction Q+ = {qt : t 2

E ) ] taken with respect to N! Q- = {qt : t I 0). Using the Markov

0). A similar expression holds for IEpo[Z$(q,

for paths in the backward direction

property of fl and the Schwarz inequality, we obtain by (3.12)

(1,q T ) 2 5 - E ~ ~ ~ ~ [ ~ - S ! T ( ~ t * e ) ( q t j) d ~l ] ~ ZT l [

. [f : ( ~~ t *- e ) ( q t ) d t ] ]

. (3.14)

Moreover, since

4

E

,-.f,T(tt*e)(qt)dt

1

= e- :J :J

W(qt-q,,t-S)dtds I

(3.15)

and by a similar expression for the [-T, 01 interval, we get

(1,\kT)’ 5

-E?o ZT l 1

= -EN0

ZT

e-fo

c

[e-

T

T

so W(qt-q,,t-s)dsdte-S_OT

sr, .rYT

W(qt - q s ,t-s)dsdt+S

s, SAT W ( q t - q s , t - s ) d s d t = EN^ [

1

Here AT = (-T,O) x (0,T) U ( 0 , T )x (-T,O). In position space the interaction potential is

./!T

SAT

W(qt-qs,t-s)dsdt

W(qt -q. , t - s ) d t d s

1

1

(3.16)

322

To show that the left hand side of (3.12) converges t o zero, we restrict to the set

with some X < 1. By using the estimate (qt - 4 s

+x

-

Y)’

+ (t

-

s ) ~ 5

8T2’

+ 2(x - Y ) +~ (t - s)~,

(3.19)

we obtain

ss,, s, .L3 dtds

e ( 4e ( 4

dx

dy (qt - qs

ds

dx

1 1 k3 L3 T

T

dt

+x - + (t y)2

-

s)2

dye(x)e(y)8T2A + 2(x -1

+ (t + s)2 (3.20)

The right hand side in (3.20) goes t o infinity as T

-+

03,

since X < 1. Hence

we have shown that

We complete the proof by showing that the limit is zero also on the complement of AT. Use again that W ( q , t )< 0 and estimate

323

As 0

< -2

ST ST -T

and

sw3dk1c12/1k12
0. This exponential growth is balanced by the second factor. We

have

as a slight modification of Lemma 5.2 in [4] based on Varadhan's lemma

taken together with the bound exp(-clx12"f1) on the decay of c > 0, f(T) is a polynomially growing correction, and a

appearing in (1.3). By taking 1/(2a!

(3.25) goes to zero with T

4

GO.

Here

> 0 is the exponent

+ 1) < X < 1, the right hand side of 0

03.

Note that Nelson's model provides the first rigorous non-trivial example disproving the widely circulating belief that Fock space-based and Euclidean quantization are equivalent.

4. The infrared convergent case: d

24

In dimensions higher than three no infrared divergence occurs and the two ways of quantizing are equivalent.

Theorem 4.1. Suppose d 2 4 and that V satisfies (Vl). T h e n f o r s u f i ciently weak couplings 0 < e < e* the Hamiltonian H has a unique strictly positive ground state @ in 7-l' at eigenvalue EO (and hence HN has a unique ground state in L2(Rd,dq) 8 3sym). Moreover, P i s absolutely continuous with respect to Po and dPdPT - lim - - lim Q $ = Q ~ . dPo

dPo

~ + m

T

-

~

324

Furthermore, there exists a unatary map

r

: 'Ti + N o such that

r d l ( H-

E0)r = He,,. We refer to the details of proof to [5]. Note that formula (4.1) provides the explicit relationship between the path measure and the ground state in Fock space. This is "failed" in the three dimensional case.

5. 3D infrared regular representation

Even though in three dimensions there is no ground state in Fock space because the Coulomb potential has too slow decay, Section 3 offers a Hilbert space and a function which can be identified as the "true" ground state. However, there is still the possibility of keeping the original framework at the cost of using another, non-equivalent representation. This will be done here. We reshuffle the field action (1.7) by adding a term dependent on a function u to be chosen below: (5.1) For convenience, we represent this auxiliary function as the convolution

and of h require that

(1)

is a real-valued, even, bounded, and sufficiently smooth function;

(2) i ( 0 )

=

1.

The first condition is more for simplicity than necessity, the second is essential. Keeping the particle's action and the total action unchanged, we obtain the reshuffled interaction

325

The function u is chosen in such a way that the free Gaussian’s mean asymptotically agrees with the solution of minimal energy of the field equation. This then leaves room for the possibility for Po to be absolutely continuous with respect to P obtained in this modified framework. The modified field is now described by the Gaussian measure

Gh

on

C(R,S’(W3)) of mean

(i.e., represented in position space in terms of the function y), and covariance

Denote its stationary measure by Gh. For convenience we will have the centred field

as an auxiliary variable; this again is distributed by a Gaussian measure

with zero mean and the same covariance as above. Note that u introduces a shift in the mean of the unmodified Gaussian measure

G on the same

space. This shift induces the unitary map : L2(S’(R3),d G h )

with F : S’(R3)

---f

+

L2(S’(R3),d G ) , ( U F ) ( f )= F ( f

-

y),

(5.7)

R, f E S‘(R3), and y given by (5.4), which we introduce

for later use. The modified non-interacting joint particle-field process on C(R, W3 x

S’(R3)) is now described by the product measure pl = N o x G h with t = 0 distribution

Pi = No x

Gh.

The interacting system has the path measure

326

with accordingly modified partition function. In this case, however, Gaussian integration gives Zh,T =

E,q Fg,,

[ exp ( - / T

-T

where 'h,T - /exp

( - /'-T x exp

/

/

i t ( k ) c ( k ) ( e i k ' q t- L ( k ) ) d k d t ) ] ]

W3

fjt(k)$(k)(eik'qt- i ( k ) ) d k d t )

W3

(

STJ' -T

(5.9)

T(k)&k)(ezk'qt - L ( k ) ) d k d t ) d Q

R3

Notice that in the above expression we passed from integration with respect to

Qh

t o integration with respect to Q. From this we get that

with CT

> 0 independent

of Q and

(5.11)

with the same W as in (2.8). This implies that the marginal distribution Nh,T

of p h , T for the particle paths satisfies

(5.12) with

2h,T

=

~ 2 ~ , d N 0Because . of picking up some extra interaction

energies, it takes some work to show that in this case too the local weak

327

limit

Nh,T -+

Nh exists as a Gibbs measure on c ( R , R 3 ) . However, this

can be done essentially in the same way as for the Gibbs measure for the unmodified system. Moreover, it turns out that i.e. it coincides with the limiting measure

Nh does not depend on h,

N constructed for h = 0. Based

on a similar argument as above that the full path measure is a mixture of a Gaussian and a Gibbsian measure, we obtain in this case too that

Lemma 5.1. There is some e* > 0 such that f o r all 0 < e 5 e* the weak local limit limT,,

ph,T = p h

C(R,R3 x S’(R3)). Moreover,

p h

exists and is a probability measure on coincides with the path measure p f o r

h = 0. In the modified function space representation Nelson’s Hamiltonian is H on the Hilbert space 3.1; = L2(R3 x S’(R3),dP;) satisfying

(5.13)

with suitable F and G. The Euclidean Hamiltonian He,, in this case, acting on the Hilbert space

‘Flh =

3.1 = L2(R3 x

is the self-

adjoint operator generating the symmetric contracting semigroup Tt given by

Theorem 5.2. Suppose the coupling 0 < e 5 e* with some e* > 0. Then

H has a unique strictly positive ground state

corresponding to its lowest

eigenvalue Eo, and H - Eo is unitary equivalent with Heuc. For a proof of the above two statements see [6]. The problem of mapping back to Fock space remains yet.

First

L2(R3,d N o ) can be mapped to L2(R3,dq) by the similarity transformation

328

R : 4(q) H 40(q)d(q). Thus the particle Hamiltonian becomes H,

=

1 --a + v(q). 2

(5.15)

For the free field U defined by (5.7) can be used first to go from

L2(S'(R3),d G h ) to L2(S'(R3), dG). This map induces a transformation between the shifted and non-shifted Gaussian processes, keeping their generator the same. Then by the Wiener-It6 transform W : L2(S'(R3),dG) +

F

this is further mapped into Fock space. Finally we transform the

interaction. By (5.13) in L2(R3 x S'(R3),dq x dGh) the interaction is multiplication by

S,, G(k)i(k) which under

(eik.q -

(5.16)

h(k)) e-ik.xdk

U goes over t o the multiplication operator

L3

(5.17)

c(k)(i(k)- q ( k ) ) ( e i k . q - h ( k ) ) e-ik.xdk

on L2(R3 x S'(R3),dq x dG). Consider now

v = (1 @ W)(1 @ U ) ( R@ 1)

(5.18)

as an isometry from L2(R3x S'(R3)),d P ) to L2(R3,dq) @ 3.Then

(5.19)

E €I$".

The standard Nelson Hamiltonian HN corresponds to setting (5.19). It is seen that subtracting by

h =

h regularizes the interaction

0 in

at small

k and makes Hr;"" a well defined Hamiltonian having a ground state in L2(R3, dq) @ F.

329

Acknowledgments: It is a pleasure to thank Herbert Spohn, Robert A. Minlos, Volker Betz and F'umio Hiroshima for joint work over the past years.

References [l]Betz, V. and Lorinczi, J.: Uniqueness of Gibbs measures relative to

Brownian motion, to appear in Ann. I.H. Poincare' B, 2003

[a] Betz,

V., Hiroshima, F., Lijrinczi, J., MinIos R.A. and Spohn, H.:

Ground state properties of the Nelson Hamiltonian: A Gibbs measurebased approach, Rev. Math. Phys. 14,173-198, 2002 [3] Lorinczi, J.: Gibbs measures for densities generated by double Itointegrals (in preparation) [4] LBrinczi, J. and Minlos, R.A.: Gibbs measures for Brownian paths

under the effect of an external and a small pair potential, J. Stat.

Phys. 105,607-649, 2001 [5] LBrinczi, J., Minlos, R.A. and Spohn, H.: The infrared behaviour in Nelson's model of a quantum particle coupled to a massless scalar field,

Ann. Henri Poincare' 3,269-295, 2002 [6] LBrinczi, J., Minlos, R.A. and Spohn, H.: Infrared regular representation of the three-dimensional massless Nelson model, Lett. Math. Phys. 59, 189-198, 2002

[7] Spohn, H.: Ground state of a quantum particle coupled to a scalar Bose field, Lett. Math. Phys. 44,9-16] 1998

Optimal filtering of stochastic parabolic equations* S. V. Lototsky

Department of Mathematics, USC, Los Angeles, CA 90089, USA email: [email protected] http://math.usc,edu/Nlototsky

Abstract An estimation problem is considered for a stochastic parabolic equation with an unknown random coefficient. The additional randomness in the coefficient generalizes a popular estimation problem that has been extensively studied in recent years. The filter estimate of the coefficient is constructed from a finite-dimensional projection of the solution of the equation. Under certain conditions this estimate is approximated using a generalized Kalman-Bucy filter whose filter variance tends to zero as the dimension of the projection increases.

Keywords. Bayes Estimator, Conditionally Gaussian Process, Degenerate

Equation, Kalman-Bucy Filter, Kushner Equation, Ricatti Equation Subjclass[2000]. Primary 60G35; Secondary 60H15, 62M20 'The work was partially supported by the Sloan Research Fellowship and by the ARO Grant DAAD 19-02- 1-0374. 330

331

1. Introduction Stochastic partial differential equations (SPDEs) are becoming more and more popular as a modelling tool in various branches of applied science. Hydrology [38, 391, mathematical finance [37], physical oceanography [7, 321, and population biology [5, 61 are some of the areas currently using SPDE-based models. Numerous other examples of applications of SPDEs can be found in the books [4, 10, 231 and in the classical paper [40]. Successful utilization of any equation as a modelling tool requires rigorous results about existence, uniqueness, and regularity properties of the solution under sufficiently general assumptions. Even though the whole topic of SPDEs is relatively young, there are already many comprehensive studies of the analytical properties of both linear and non-linear equations, for examples, [lo, 23, 24, 34, 35, 401. Still, these analytic results are only the first step toward efficient practical use of SPDEs. Indeed, every time a real-life process is represented by an equation, only the general form of the equation is known and the details must be determined by reconciling the model with the observations of the process. In other words, an inverse problem must be solved to find, on the basis of the observations, the coefficients, free terms, and, sometimes, initial and boundary conditions in the equation. For stochastic equations, inverse problems are usually solved by methods of statistical inference, using the observations as the input of a suitable estimator. The key mathematical question is the asymptotic behavior of the estimator, that is, whether the estimator approaches the true value, and how fast, as more and more of the observations become available. In particular, long time asymptotic assumes increase of the observation time, and small noise asymptotic, decrease of the noise intensity in the equation. The first works on statistical inference for SPDEs [l,2, 271 studied esti-

332

mators in the long time asymptotic. Later, models with small observation noise were introduced and studied [8, 12, 18, 19, 20, 211. While some of the papers address estimation of the initial condition and free force, the majority of the research has been on parameter estimation. The following model has become especially popular:

du(t,IC)

=

( d o + 6 ( t ,IC)d1)u(t1x)dt + ~ d W ( IC), t , 0 < t 5 TI ' ~ l l t = , ~U O , (1.1)

where 6 = 6 ( t ,x) is an unknown coefficient, d o and tial operators so that d o

d1

are known differen-

+ 6d1 is elliptic for all admissible values of 6, and

W is a space-time white noise. The equation is considered on a smooth compact manifold or in a smooth bounded domain with some boundary conditions. Long time asymptotic means T

+ 03,

and small noise, E

4

0.

Under certain conditions, a consistent estimator of 6 in (1.1)is possible even if T and

E

are fixed. Any computable estimator must be based on a

finite dimensional projection of the solution of (1.1), and, if the order of the operator

d1

is sufficiently high, then the estimator can approach the

true value as the dimension of the projection increases even if the observation time and the noise intensity remain fixed. This asymptotic behavior, known as spectral asymptotic, is much more interesting than either long time or small noise, and has no analogs in finite-dimensional setting. The projection-based, or spectral, estimators were first introduced in [13], and further studied in [17], for the model of the type (1.1) with one unknown scalar parameter O ( t , x ) = 00 E R and with commuting operators d o , dl. The commutativity assumption ensures that the SPDE (1.1)is diagonalizable, that is, can be reduced to a system of uncoupled ordinary differential equations. Further work in that direction included analysis of maximum likelihood-type estimators for several scalar coefficients [ll], sieve and kernel estimators for time-dependent coefficients 0 = 0 ( t ) [14, 151, and Bayes-type

333

estimators [3]. Non-diagonalizable models were also studied [16, 301. In all the above works, the unknown coefficient 0 was assumed deterministic. Additional randomness in 0 not only makes the model more general but also poses new and interesting mathematical challenges. When 0

= 0(t)

is random, the corresponding estimation problem becomes the problem of filtering, with the solution u of the SPDE (1.1) being the observation process. There are two classical filtering models, linear Gaussian and nonlinear diffusion, that have been extensively studied from both theoretical and applied points of view. Filtering problem for equation (1.1) does not fall into either of the categories. Indeed, to have existence and uniqueness of solution of (l,l),the unknown process 6 must be uniformly bounded and therefore modelled by a nonlinear diffusion equation with degenerating coefficients, while the right-hand side of the observation process u is linear in

u.As a result, new constructions and technical tools are necessary to carry out the analysis. In the current paper, the filtering problem is studied for equation (1.1) with

E

= 1, fixed T

> 0, and a diffusion process 0

= 0 ( t ) as the un-

known coefficient. The underlying SPDE is assumed diagonalizable, and a finite-dimensional projection of the solution represents the observation process. The unknown coefficient is modelled by an Ito equation with coefficients degenerating at the end points of some interval. The special form of degeneracy ensures that the process never leaves the interval, while the corresponding filtering equations have a unique solution in a certain weighted function space. The filtering density satisfies a non-linear Kushner-type equation, and admits an alternative representation as a normalized solution of a linear Zakai-type equation. Under certain assumptions, an approximation of the optimal filter is constructed using a generalization of the Kalman-Bucy filter. The same condition on the order of the operators

334

as in [17] ensures that, for every T

> 0, the solution of the correspond-

ing Ricatti equation, representing the filter variance, tends to zero as the dimension of the observation process increases. Section 2 presents the basic existence, uniqueness, and regularity results for equation (1.1) under the assumption that the process 8 is compactly supported. The filtering problem is studied in Section 3. The main result is that the filtering density exists and is a smooth function with the same support as 8. The proof is based on recent results about solvability of degenerate parabolic equations in domains [29]. The spectral asymptotic of the filter is studied in Section 4 when 8 can be approximated, in a certain sense, by a Gaussian process.

2. Stochastic Parabolic Equations With Random

Coefficients Let G be either a smooth bounded domain in Rd or a smooth compact ddimensional manifold without boundary and with a smooth positive measure. Denote by C r (G) the collection of infinitely differentiable, compactly supported, complex-valued functions on G. Let do and dl be differential or pseudo-differential operators on CF(G). If G is a bounded domain, then, to simplify the presentation, all operators will be considered with zero boundary conditions. On a stochastic basis

F

=

(R,.F,{Ft}t,o,P) with the usual assump-

tions (see, for example, [22]), consider a cylindrical Brownian motion

W = W ( ~ , Z ) .In other words, W is a random process with values in the set D'(G) of distributions on G so that, for every

'p E

C F ( G ) with

II'pIIL2(c) = 1, (W,'p)(t) is a standard Wiener process on IF, and for all (Pl,(P2

E c r ( G ) ,E(W,$91)(t)(wp2)(s)

=min(t,s)

'

('P1792)Lz(G).

For a predictable random process 6 = 8 ( t ) on IF and a D'(G)-valued

335

random variable uo,consider the following equation:

u(0,Z) = uo(2).

Definition 2.1. A predictable process u with values in D'(G) is called a solution of (2.1) if and only if, for every 'p E Cr(G), the equality

(u,'p)(t) = (UOl'p)

+

1

1 t

t

(.Abl.)(.)dS

+

+ (W,'p)(t)

~(s)(4'p,u)(s)ds

holds with probability one for all t E [0,T ] at once, where d: is the formal adjoint of di, that is, the operator so that

Remark 2.1. It is possible to consider a more general noise process W with a correlation operator 8. As long as 8 is invertible, the corresponding equation is reduced to (2.1) by applying the operator 8-' to every term.

Definition 2.2. Equation (2.1) is called diagonalizable if and only if the following conditions hold:

D1 There is a complete orthonormal system {hk, k 2 l} in &(G) so that

D2 There exist positive finite limits

where mi is the order of the operator di.

D 3 There exist positive real numbers and w E 0,

where 2m = max(m0, ml).

C ~ , Q so

that, for all t E [O,T]

336

Conditions D1-D3 hold in many physical models (see, for example, [32,

331). A typical situation is when the operators do and either do or

d1

d1

commute and

is uniformly elliptic and formally self-adjoint. More details

can be found in [36].

cF(G)define ' p k = s, ' p ( x ) h k ( z ) d z . Conditions D1-D3 imply that, for every n > 0, there exists a c = c('p, n ) > 0 so that l'pkl 5 Ck-". For

'p E

Therefore, for every y E R, we can define the norm

11 . (IY

on CF(G) as

follows: llP# =

c

k2yldl'pk12.

k>l

(1 . (Iy. An

Let H Y be the completion of CF(G) with respect to the norm

element w E HY is then identified with a sequence {Wk, k 2 1) of complex numbers so that

11~113= c k > l k2Yldlv12 < -

Fourier coefficients of w, and v E

HY

00.

The numbers

Wk

are called

is represented as a formal Fourier

series w(z) = Ck,l wkhk(z). -

Remark 2.2. A similar construction of the Hilbert spaces H Y can be carried out when equation (2.1) is not diagonalizable; see [30]for details.

Theorem 2.3. Assume that equation (2.1) i s diagonalizable and uo E

L,(R; H') f o r some r < - d / 2 . T h e n there is a unique solution u of (2.1). This solution belongs to the space

&(R x (0, T ) ;Hm+') n L z ( R ; C((O,T),H')) and satisfies

where m is f r o m condition D3.

Proof. It is know (see, for example, [24])that W ( t , x )=

ck,lw ( t ) -

hk(z),where wk, k 2 1, are independent Wiener processes. As a result, for every r

< - d / 2 , the process W is an H'-valued continuous square

integrable martingale with quadratic variation ( W ) ,= t

k 2'ld.

337

Next, conditions D1-D3 imply that, for every T E R, there exist positive numbers C1, C2 so that, for all

'p

E CF(G), t E

%(((A + e ( t ) d l ) ~Y ,) , )

[O,T],and w

+ c1IIpII:+m 5 2'

E

0,

IIYII:,

(2.3)

where %(.) is the real part of the expression. Indeed, it follows from (2.2) that

(1) There exists a positive number

[O,TI, w

E

a1

k 2 1 and all t

so that, all

E

0,

(2) There exists an integer ko and a positive number

a2

so that, for all

k > ko and all t E [O,T],w E R,

5

-a2

c

Iqpk2('+")'d

+ (a1 + a2)k,2"/d

k> 1 =

c

lYk12k2'/d

k>l

- ~ ~ I I Y I I 2, + ~ + (a1 + a 2 ) k o2m/d IIYIIP.

The statement of the theorem now follows from Theorem 3.1.4 and Remark 3.4.9 in [35].

17

The Fourier coefficients 'ZLk = u k ( t ) , k 2 1, of the solution of (2.1) satisfy the following uncoupled system of stochastic ordinary differential equations

338

As a result, under conditions Dl-D3, the infinite collection of equations (2.4) is equivalent to (2.1), and the solution of (2.1) can be written as a Fourier series

uk(t)hk(x),

u(t,x) =

(2.5)

k> 1

converging in the corresponding Hilbert space. Note that condition D1 implies equations (2.4) for the Fourier coefficients of u,while conditions D2 and D3 ensure the appropriate convergence of the Fourier series (2.5).

Example. The following equation is a one-dimensional version of the heat balance equation from physical oceanography [7, 321:

du(t,x) = (uz, - e(t)uz(t, x) - u ( t ,x))dt+ d y t , x), o < t 5 T , o

< < 1, (2.6)

with periodic boundary conditions. In this example, G d2

do=

- 1,

d

A1 = --z,so that

mo = 2m = 2, ml

ordering, hk(x) = eanikX,where i = Vk =

=

m,so that

=

S1, a circle,

1. With a suitable

Kk

= -47r2k2

-

1,

-27rik, and condition (2.2) holds as long as there exist real number

ae, be so that O ( t ) E [as,be] for all t E [0, T ] and w

E

52.

Alternatively, one can consider

+

d q t , x) = (e(t)u,x- u,(t, x) - u(t,z))dt ~ ( x), t ,o < t

I T , o < < 1, (2.7)

with periodic boundary conditions. Then do =

-zd - 1, A1

=

=, d2 so

that mo = 1, m l = 2. In this case, condition (2.2) holds as long as there exist positive numbers

a0

and be so that e(t) E [as,be] for all t E [0,TI and

w E 52.

0 Remark 2.4. While it is possible to construct a path-wise solution of equation (2.1) if the numbers C1 and

C2

in condition (2.3) are random and

339

not uniformly bounded, this path-wise construction complicates the further analysis and is not discussed in the current paper.

3. Optimal Nonlinear Filtering of Diagonalizable Equations Consider the problem of estimating the random process 6' from the observations of the first N Fourier coefficients (2.4) of the solution of the diagonalizable equation (2.1). The solution of this estimation problem is, of course, impossible without additional assumptions about the process 6'. Condition (2.2) implies that the process 6' is uniformly bounded: there exist real numbers ae, be so that, for all w E R, inf e(t) 2 ae, sup e(t) I be. OltST

OStlT

If dl is not the leading operator, that is, if ml < mo = 2m, then there are no further restrictions on the numbers ae, be. If d1 is the leading operator, that is, mo

< ml

= 2m, then (2.2) implies as

> 0, that is, 6' must be uni-

formly positive. Finally, if mo = ml = 2m, then the bounds on ae, be will be determined by the asymptotic behavior of the sequences k-2mldvk and k-2mldKk;

this asymptotic behavior depends on the particular operators

(see [36] for details).

A possible model for 6' is the Ito diffusion equation: dO(t) = B ( t ,e ( t ) ) d t

+ r ( t ,B ( t ) ) d V ( t ) ,

(34

where B and r are sufficiently regular functions and, for simplicity] the Wiener process V is independent of W . The initial choice of the functions

B and r might not guarantee that (3.1) holds. Below is a general procedure for modifying equation (3.2) to ensure that condition (3.1) holds. Let p = p ( x ) be a smooth, compactly supported function on R so that

340

(1) There exist finite nonzero limits

Iim p ( x ) / ( x- ae) and

x+ag

lim p ( x ) / ( b e

x+be

-

x);

(2) p ( x ) > 0 for x E (ae,be); (3) p ( x ) = 1 on [as

+ 15, be

-

61 for some sufficiently small S > 0.

Such a function exists and can be constructed by appropriately mollifying

be]. Note that p(ae) = p ( b e ) = the characteristic function of the interval [a@, 0 , while the first derivative of p at those points is not zero. Consider the following modification of equation (3.2):

with some initial condition 0 0 , independent of V and W.

Proposition 3.1. Assume that, for 0 5 t 5 T and x E [ae,be], the func-

tions B = B ( t ,x ) and r = r ( t ,x ) are deterministic, bounded, and Lipschitz continuous in x , uniformly in (t,x ) . Then equation (3.3) has a unique strong solution for every square-integrable initial condition. If the initial condition 00 is a random variable whose distribution is supported in [a@, be], then the solution of (3.3) satisfies (3.1).

Proof. Recall that p is a smooth compactly supported function. Assumptions about B and r imply that the coefficients in (3.3) are bounded and uniformly Lipschitz continuous in 2. The first statement of the proposition is then a consequence of the general solvability theorem for the Ito equations (see, for example, Theorem 5.2.1 in [31]). The second statement of the proposition is the consequence of the uniqueness of solution of (3.3). Indeed, by assumption on the function p, p(ae) = p(be) = 0 , which means that the constant functions O ( t ) = ae and O ( t ) = be satisfy (3.3). Therefore, each solution of (3.3) starting in [as,bs] will stay in that interval for all t > 0.

0

341

The above proposition shows that (3.3) is an acceptable model of the coefficient process 8. The filtering problem can now be stated for the unobserved state process 8 and the observation process ~

. . ,U N :

1 , .

The filtering problem for (3.4) consists in computing the conditional density of 8 ( t ) given the observations up to time t. It is known [35, Chapter 61 that, under certain regularity assumptions, the conditional density in the diffusion filtering model satisfies a nonlinear stochastic parabolic equation, also know as Kushner’s equation. Alternatively, the density can be computed by normalizing the solution of the linear Zakai equation. While the usual regularity assumptions for diffusion filtering models are not satisfied for (3.4), the corresponding equations can still be derived and studied. Recall that the filtering density for (3.4) is a random field II = II(t,x) so that, for every bounded measurable function F = F ( x ) ,

q F ( e ( t ) ) l u k ( s )IC, = 1,.. . , N ; o < 5 t ) = Under condition (3.1), it is natural to expect

s,

f(x)n(t,x)dx.

II to be supported in [ae,be]

for all t. Let

be the generator of 8. If the functions B and r are sufficiently smooth in

x , then the adjoint C* of L is defined by

a2

d 1 ( L * f ) ( t , x= ) --&( P ( x ) B ( t , x ) f ( x+ ) )2%

( p 2 ( x ) r 2 ( t , x(4) )f .

Theorem 3.2. Let the following conditions be fulfilled: 1. The functions B and r are infinitely differentiable in x on [a@, be] so

342

that each derivative with respect to x i s uniformly bounded as a function of

t and x . 2. There exists a n E > 0 so that r2(t,x ) 2 E for all t E [0,TI and x E [ae,be]. 3. The Wiener process V i s independent of W .

4. The initial condition 00 i s independent

of V and W and has a density

no E Co"((ae,be)). T h e n the filtering density IT = I I ( t , x ) for (3.4) exists and has the following properties: (1) For every t E [0,TI and P-a.a. w E 0 , the support of T1 i s [ae,be]

and the function IT i s infinitely differentiable with respect t o x with all the derivatives vanishing at points ae and be. (2) The function I2 is a path-wise solution of the non-linear equation

with initial condition ITo, where Bk(t)= JRd(tck

+xvk)n(t,x)dx.

Proof. While (3.5) is the formal Kushner equation for (3.4), the coefficients in (3.4) do not satisfy several technical assumptions that are traditionally used in the literature to derive the equation and study its properties. Specifically, the operator L is not uniformly elliptic (because of the function p ) and the observation functions H k ( x , y ) =

(tck

+ v k x ) y are not

bounded in either x or y. This difficulty is resolved by using approximations and certain results about solvability of stochastic parabolic equations in weighted spaces.

343

For M = 1,2,.

. ., define the stopping time

if S U ~ ~ p, y ~ ( t 0 5 ) to). By Lemma 4.4 we conclude that limN+CO to)

=0

with probability one. If Ib(t)12 2

by Lemma 4.3, there exist non-negative functions

E

> 0 for all t , then,

YN, i j ~ both ,

satisfying

equations of the type (4.8), and a positive integer-valued random variable p satisfying P(p < CQ) = 1 so that, for all N By Lemma 4.4 we conclude that limN+m

> p, i j ~ ( t 0 5 ) Y N ( ~ OI ) YN(~o). $ N Y N ( ~ O )exists

and is positive.

Theorem 4.2 is proved. Theorem 4.2 shows that, under condition (4.6), the variance of the linear filter tends to zero as more and more of the spatial Fourier coefficients of the solution of (2.1) are included in the observation process. The question remains open whether a similar result holds for the non-linear filter. In the example at the end of Section 2, condition (4.6) holds for both equations; for equation du = Au-a. Vu+O(t)uwith a known vector

a, this condition

349

holds if and only if d 2 2 [17]. Equations (4.4) can be solved explicitly if a ( t ) = b ( t ) = 0. In that case,

6 ( t ) = O0 and ~ N ( T )=

mo

+

N

70 C k = l

1

where mo =

IE(&),

+

so T

70

V k W ( t ) ( d U k ( t )- K k % ( t ) ) d t

z:=1:J

1

.,".",t)dt

(4.9)

70 = war(&), and uo is assumed deterministic. If

(4.6) holds, then direct computations show that limN,,

B^N(T) = 00 with

probability one. Similarly, limN-,m $ ~ Y N ( T has ) a positive finite limit with probability one, which does not contradict Theorem 4.2. For a constant coefficient 6, expression (4.9) is an example of the Bayes estimator. It is known from [17] that, if 6 ( t ) = 6, a real number, then the maximum

iNof 0 is given by

likelihood estimator

e^N =

cr='=, s,' c:=,s,'44 VkUk(t)(duk(t)

- 6kuk(t))dt

By comparing (4.9) with (4.10), we conclude that, for every a limN,,$E(6N(T)

-

(4.10)

(Wt

i N =) 0 with probability

one and in every


I{j€{k,k+l,.’.

,2n}:

E {0,1}7 determines exactly one non-crossing

{ I h , ~h}:,~,

over the set {1,2,. . . ,2n} in the sense of

{zh}:=l={jE{l,...

,2n}: E = 0 }

Where, just by renaming, one (and only one) of the two sets {~h}:,~

set { Ih}:=l

E=O}I

{Ih}:=l

and

can be assumed to be an ordered set and we shall suppose that the has increasing order. Moreover, the non-crossing pair partitions

which satisfy

= 2 n play a special rule in our investigation and we shall

call them, as in [2, 3, 41, totally-connected (for simplicity t.c.) non-crossing pair partition. In [5],it is proved that for any

E

E (0, l}?, the scalar product (2.8) can

be written as

(2.9) where, m and d l , . . . ,d, are determined uniquely by the E E (0, l}?. This result is named the factorization principle.

360

By the factorization principle, in order to understand (2.9), the problem is reduced to know it for such

E

E {0,1}?

that determines a t.c. non-

crossing pair partition. jFrom now, we shall restrict ourselves to the case M = [O,T]and A, =

xo,

Vn E

N.

Lemma 2.2. For any n E N,

fl,.

..f i n

E

H and E

E (0,

l}?

being totally

connected,

Proof. Let us prove this Lemma by induction on n. If n = 1, Lemma 2.2 is obvious. In the case n = 2, (2.10) becomes

(2.12) becomes to

So the thesis is verified. Suppose that the conclusion of Lemma is true for n - 1, let's examine the conclusion for n. Since we are considering the totally connected case, r1 must be equal to

2n. In this case, 2n-1 can not be a left-index

lh

if n 2 2 (since if Zh

= 2n-1,

361

then 2n must be equal t o

~ h ) . By

the assumption that the non-crossing

pair partition is totally connected, we know that But it is certainly true that

11

TI

= 2n = rh, i.e. h = 1.

= 1 and this shows that n = 1 which is

contrary to n 2 2. The first case to be considered is 1, = 2n - 2. In this case,

By Lemma 2.1, we know that .(fin-2

+ ).

( f 2 n - 1+ .)

(fin @

=

lT

dY(72,-2

f 2 n - 1 ) (!I). ( f+ i n X [ 0 , 4) @

(2.14)

then the quantity in (2.13) becomes LTdy(72n-2f2n-1)(?/) (@?.(fl)

'

"~+(f2n-3)~+(finX[O,y))@)

This quantity is equal, by the induction assumption, t o

Jlu

T

dY(72%2 f i n - 1) (Y)

s' 0

dJ:(71f2n X [ O ,y)1(J:) .~(2n-3) ( f 2 n - - 3 X ( z , T ] ) @ )

(2.15) By exchanging the order of integrals and using the fact

and this is exactly what we want to prove.

X[O,~)(IC)=

362

Another case to be considered is that 1, crossing principle, we know that

T,

= 1,

< 2n - 2. Thanks to the non-

+ 1 and any j

2 1,

+ 1 is a right

index, i.e. ~ ( j=) 1. So .E(zn)(fzn)aE(T,)(fr,). . . u E ( 2 n ) ( f 2 n )= @ a ( f z n ) a + ( f ~ , + lU)+~ ( f 2~, )~@ =

lT

dY(finfT")(Y)a+(fr,+lXIO,y))

.'.a+(f2nx[o,y))@

By substituting this formula, the left hand side of (2.10) becomes

'

(fZ,-lX(z,T])a(rnfl)(f~,+lX(z,T])

' ' '

(2,- 1 )

x

( f 2 n- 1 (z ,T]

@)

(2.18) by Lemma 2.1, the quantity (2.18) is equal to

1

T d s ( T 1 f 2 n ) ( 2 )( @ I ~ " ( ~ ) ( ~ ~ 'X "~ ( Z E (, 2T"]- )1 ) ( f 2 n - ~ x ( 2 , T ] ) @(2.19) )

Therefore, the proof is completed.

363

3. The distribution of field operator

Now let us consider the vacuum distribution of the backward field operator,

+

~]) for [ E namely, the distribution of the operator a ( t ~ ( ~ , a+([X(t,q) L2 ( [ O , TI) and t E [0,TI. Since in our case, the field operator is bounded, the distribution is determined uniquely by its moments. Therefore, we shall first calculate the moments of the field operator with respect to the vacuum state. For each

E

E L 2 ( [ 0TI) , and rn E

+

N , just by expanding [ a ( [ ) a+(E)]"

to a sum of products of m creation and annihilation operators, we know immediately that, with respect to the vacuum state, all odd moments of field operator are equal to zero. For each t E [O,T],n E N and a continuous function on [O,T],say

t,

let's define

It is clear that W C , ~ (isT )equal zero for all n E N . In the following, we shall assume that t E [O,T). Let's denote qt :=

st' IEI2 (s)ds .

It is easy to see that vc,o(t) =
= 1 and

=

1'

I E 1 2 ( S ) X ( t , T ] ( S )=

I'

IE12((.)

= qt

(3.2)

+

For any n 2 2, we expand [ a ( t ~ ( ~ , T la) + ( [ ~ ( ~ , into ~ l )a] sum ~ ~ of products of 2n creation and annihilation operators: (3.3)

364

We know that many terms in (3.3) have vacuum expectation value zero. More precisely,

Theorem 3.1. For a n y n E N , t E [O,T],the sequence of functions {v+(t)}F=l satisfies the system of diference-differential equation

c

ut,n+l(t) =

vt,n-k(t)

k=O

Proof As we have seen, each crossing pair partitions

{Zh,

/'

T

n

E

E

~SIEI~(S)~~,~(S)

t

(0, l}? determines exactly one non-

T ~ } E ,on~ the set ( 1 , 2 , . . ,2n}.

non-crossing principle we know that

(3.5)

9

TI

E

Thanks to the

{ 2 , 4 , . . . ,2n}. Now we split the

set (0, I}? into n-parts according to all possible values of the first right index

r1:

(0, I}? =

n

n

k=l

k=l

U(0, l } F k := U

{E

E

10,

: r1 = 21~)

and find that

By the factorization principle, the right hand side of (3.6) is equal to n+l

By Lemma 2.2, the first scalar product in (3.7) is equal t o

365

1

T

=

(

dslC12(s) @la'(2)(~X(s,T]) ' ' 'a'(2k-1)(EX(s,T])@)

For each k E { 1 , 2 , . . . , n

+ l}, as E runs over {0,1}";12,

(3.8)

its restriction on

the set { 2 k + l , 2k+2, .. . ,2 n + 2 } runs over all non-crossing pair partitions on the same set. Similarly, as

E

runs over ( 0 , 1}EL2, its restriction on the

set { 2 , 3 , . . . , 2 k - l}gives the totality of all non-crossing pair partitions on this set. Thus, we find that

n

k=O

,T

(3.9)

Jt

Therefore the thesis is finished.

Now we investigate the generating function of { w ~ , n ( t ) } ~ = o

n=O

for x around the origin. By the same arguments as in [ 5 ] ,we know that this generating function is well defined on a non-trivial interval including the origin. Moreover, we have the following

Theorem 3.2. For any t E [O,T],in its interval of convergence, the gen-

erating function RE@,x) i s equal to 1 J-.

Proof Just by repeating the same arguments as the proof of Lemma (3.5) of [ 5 ] ,we know that the generating function REsatisfying the integral equation

366

It is easy t o show that the equation (3.10) has unique solution which is equal to

Now we are ready to represent our main results for the distribution of the backward and forward field operator.

Theorem 3.3. For any t E [O,T],the random variables

are absolutely continuous and the density function of bt i s equal to

G ~ ( x:= ) T

1

Jmx(-@7@)

(3.11)

(x)

and the density function of bt,c i s given by (3.12)

where,

In particular, if t < T < 00, the random variable

has the density function 1

G t ( x ):= n&(T

-

t ) - x2 X-(&TZJ,&TT-))

(x)

(3.13)

367

and for any t > 0, the random variable

has the density function (3.14)

Proof (3.13) (resp. (3.14)) is obviously a direct conclusion of (3.11) (resp. (3.12)) and therefore we give only the proof of (3.11) and (3.12). Since the random variable bi is bounded, its distribution is determined by its moments, i.e. by its generating function around the origin. Therefore, in order to prove the theorem, it is sufficient to show that the generating function determined by the density function G! ( x ) is the same as RE( t ,x ) at least for x runs over a small interval around the origin.

It is obvious that for any n E N /zZn-'G;(z)dr = 0 Let's denote

(3.15)

dn(t):= / x Z n G ; ( x ) d x what we must prove is that

(3.16) In fact,

c oc)

co

yndn(t)=

n=O

00

y n /x2nGc,t(x)dx=

1 yn

n=O

(3.17)

By change of variable u := A we have

@'

X=U&

,

dx=J2rltdu

368

and the integral in the right hand side of (3.17) becomes (3.18)

which is equal to

This shows that the quantity in (3.17) is equal to (an - I)(2n - 3 ) . ' . . 5 . 3

-

n!

n=l

1

4=6

(3.20)

where, the equality in (3.20) is nothing else but the Taylor's formula. Let's now consider the distribution of the forward field operator bt,c =

+

a(Jx[o,t)) a+(Jxp,t)). We denote

Jt

:=

J . X p t ) which is in L 2 ( [ 0 , T ] ) .

By the above proof, we know that the distribution of bt,c is given by the density function

1 7rd2

6IJt12(s)ds

By the definition of

-

x2X

(-dzJT

I E t lz(s)dss+

JT

(3.21) IEtlZ(s)ds)

It,it is obvious that

Therefore we finish the proof.

References [l]L. Acccardi, Y.G. Lu: The Wigner semi-circle law in quantum electrodynamics, Commun. Math. Phys. 180, 605-632 (1996). [2] M. DeGiosa, Y.G. Lu: The free creation and annihilation operators as

the central limit of quantum Bernoulli process, Random Operators

and Stoch. Equ. Vol. 3, No. 2, pp.19-34, (1997).

369

[3] M. DeGiosa, Y.G. Lu: q-creation and annihilation operators on Inter-

acting Fock space, Japanese Journal of Mathematics, Vol. 24, No.1 pp.149-167, (1998). [4] Y.G. Lu: On Interacting Free Fock space and Deformed Wigner Law,

J. Nagoya Math. Vo1.145, pp.1-28, (1997). [5] Y.G. Lu: An Interacting Free Fock Space and the Arcsine Law, Proba.

Th. Math. Stat. Vol. 4, pp.15-32, (1997). [6] Y.G. Lu: A remark on the free central limit theorem, Bollentino di

UMI, (7) 11-B, pp.267-306, (1997). [7] Y.G. Lu: Interacting Fock space related to the Anderson model. Inf.

Dim. Ana. Quantum Proba., Vol.1, No.2 pp.247-283, (1998)

Stochastic holonomy Itaru Mitoma Department of Mathematics, Saga Wniversity,Saga, 840-8502, Japan e-mail: mitomaQms. saga-u.ac.jp

1. Introduction Inspired by S.Albeverio and his colleagues[l,2], we have been studying Chern-Simons theory from the view point of the infinite dimensional stochastic analysis [16,171. Let M be a compact oriented 3-dimensional manifold

,4 a skew

symmetric matrix algebra which is a linear space with the inner product

( X , Y ) = T r X Y * = - T r X Y , where T r denotes the trace and A the collection of all 4-valued 1-forms. Then the Chern-Simons integral of F ( A ) is equal to

F ( A )eL(A)2)(A ) , where

::s,

L ( A ) = --

Tr{AAdA+ :AAAI\A}.

Among the various integrands, the most typical example of gauge invariant observables is the Wilson line such that F ( A ) = y j , j = 1 , 2 , ...s are closed oriented loops. 370

n,”=,TrPeS7j

A,

where

371

To handle this in an abstract Wiener space setting, we need to extend the holonomy

from smooth A to rough A , which is suitable for the abstract Wiener space setting . We consider the perturbative formulation of the Chern-Simons integral

[4,5,10]. After the heuristic argument of introducing external fermionic forms and bosonic 3-form

4

and changing the Lagrangian L ( A ) accord-

ing to the method of super fields and finally integrating by the fermionic Feynman measure, the remaining terms that we expect to give the mathematical meanings are written in the variables ( A ,4 ) and the quadratic form

J M T r A A d Ais changed to ( ( A , ~ ) , Q A ~ ( A where ,~))+ (.,.)+ , denotes the inner product of the Hilbert space L2(R+) = L2(R1 @ R3) induced from

(., .)o. Here R" is the space of S-valued n-forms with the inner product

and the norm

where

*

where

J4

is the Hodge *-operation. Further Q A ~= & ( * d ~ , =

-4

a two form and

if

4

is a zero or a three form,

J4

=

4

if

4

+ ~A,*)J,

is a one or

is the covariant exterior derivative such that A0 is an

isolated critical point of the Lagrangian L ( A ) ,d~~ = d+[Ao, .] and [ A ,B ] =

Ci,i[EiEj- EjEi]Air\ B j , if A =

d

A " . E, and B = C",=,B" . E,,

where Eu,u = 1 , 2 , ...,d are the orthonormal basis of G. Then further constraint for the extension is to realize it in the framework of

Z

= ( A ,4 ) = ( Z A , X+) E Q1@ R3.

372

The aim of this paper is to discuss the extension and examine the integrabilities for the analytic functions in the sense of P.Malliavin and S.Taniguchi[l5]. The outline is based on N. Ikeda and Y. Ochi’s idea[ll,l9] of considering the line integral as the random current and the Chen iterated

integrals expression of the holonomy.

2. Definitions and Results First of all, for the simplicity we assume ASSUMPTION. All order De Rham cohomology

H * ( M ,d A o ) = 0. For any non-negative integer b and for any a,P E R”, define the inner product

( ( a P, ) ) b = ( a ,&:;P)o.

Hb(a+) denotes the Hilbert subspace of L2(R+) with the inner product ( ( ( A 4)i , ( B ,‘ P ) ) ) b = ( ( A ,$1, QY0((B, ’P))o. Define the &norm by

11 1: ‘

=

’))b.

We note

Since

Q A ~is

self-adjoint and elliptic, Q A has ~ pure point spectrum [9].

Let p i , ei = (ef =

zf=l efYA . Et,

ef =

zf=l . E t ) ,i e;”

= 1 , 2 , . . . be the

. the ASUUMPTION, the eigenvectors form eigenvalues and vectors of Q A ~ By a CONS(Comp1ete Orthonormal System) of L2(R+).

3 73

Let

fin be the set of all real valued n-forms and

Noticing that X A =

2

=

d

xz . E, and

x+ =

d Cu=l x;

. E,, we define for

(x1,x2,... , x ~ ) ~= x ”(xz,x;) and j j = (y1,y2,... , y d ) , y u = (yg,yz)

and

Then the real Hilbert space of 2 corresponding to Hb

Hb(Q+)

is denoted by

with the inner product M

j=1

and b-norm by

Now we proceed t o an extension of the holonomy t o the stochastic holonomy. Let A be a 4-valued smooth 1-form and y(t) : t € [ O , 11

-

M a smooth

closed curve in M. Recall d

A=

C A“ . E,. u=l

Let U be a coordinate neighborhood such that a neighborhood of {y(s), tl 5 s

5 t z } is contained in U.

374

In U, we have

(2.3) In the sequel we denote by

Cb,

b = 1,2, ... the constants.

By the Sobolev lemma and the bundle version of the Gaffney-Garding inequality (Theorem 7.2.6[18], see also page 44 in [9] and (iii) of Theorem 5.2 in [13]), we have a positive integer q such that the right hand side of (2.3) is dominated by

(2.1) implies On the other hand, the ASSUMPTION

which,together with

yields

By (2.3) and (2.4), we have C$(t)E H-,(R1) such that

3 75

and

Ilc:(t)

- c:(.)ll-q

5 c5 I t - s I .

(2.5)

/--.---

By the Riesz theorem, there exists Cz(t)in Hq(Q1)satisfying

where

H-b(Q1) is

the dual space of

canonical bilinear form on Let p

Hb(Q’)

Hb(Q1)and b

< ., . > - b denotes the

x H-b(Q1).

> q. Then if A E H p ( Q 1 ) ,

-

( ( A c:(t))I4 , =q< A, c:(t)>-q=

< A, cE(t)> - p =

-

( ( A C,”(t))),, ,

so that we get

which implies

For A E Hp(Q’), by (2.2), the righthand side of (2.6) is equal t o

-

-

( ( ( A4, ) ,(C,”(t),0 ) ) ) P = (C C W P ,

A

(2.7)

-

where C,”(t)= (C{(t),0). Therefore

By Chen’s iterated integrals (Theorem 4.3 in

5 1.4 of [7],p31,see also [20]),

we have

Now we will find an extension of the holonomy in an abstract Wiener space setting.

376

Let H

=

H p and (B,H , p ) the abstract Wiener space such that p is a

Gaussian measure satisfying

< 2 , > denotes the canonical bilinear form on B x (B)*. We denote ( d l z 2 , . . ,.z d ) E B by z, (E1,E2,... ,td)E B* by E , the norm of B by

where

1) JIB

and the dual norm by

11 . IIp.

By the nuclearity of the system of semi-norms { 1) . Ilb, b = 1 , 2 , .. . } guaranteed by the increasig rates of eigenvalues of

Q A ~

( (c) of Lemma 1.6.3 in [9]), there exists some integer po independent of p

-

such that B is realized as H--p-po [S].

-

If we choose sufficiently large p such that p > po

+ q , then CZ(t) E

Hpfpo(fll)and hence (CZ(t),O)E B*. Henceforth, we take this suitable space as B throughout this paper. Along (2.7), for any

5

E

B, define d u=l

where

Then also along (2.8), we define

r=l

where

wy(4=

I’ l1

...

Since S

F(& + A ) = j=1

TrPeJy j

AofA I

377

we define S

FA0

TTW7j (.).

(x)= j=1

Then we shall see the well definedness, the C" in H-Frkchet differentiation and the integrabilities, ((3) of THEOREM), for the analytic functions in the sense of P.Mal1iavin and S.Taniguchi[l5]. Define

and for the n x n matrix E

=

(q), define n

Then we have

TEOREM. (1) IVY(.) is well defined and C" in H-Frkhet differentiation.

(2) For any natural number b, we have

(3) For every positive real number

c O3

k=l

Sk

ZE[(

c

s,

llDkW7(5)(&,i 2 , ... &)l12)bl~ < +m 1

Kl ,K2 ,_ .. ,h;,

and

3.

Proof

Define d

C E = max u=l

I I

IIEull and A, ( t )=

I,,,

At.

378

Since by (2.5),

so that set

(T

= dcScE, the above term

which implies the well definedness and C"- Frkchet differentiability in H . The following lemma is known as the Fernique theorem[l2]. LEMMA2. There exists 6 > 0, such that

Since by (3.1),

Lemma 2 and (3.2) yield the proof of (2) of THEOREM. Now we proceed to checking (3).

379

Setting W j ( z )= W{(z), we have

Similarly we use the Schwarz inequality recursively, the last term

In the sequel Ccombll,lz,,,,,lk denotes the summation taken over all combinations with order of

11,/2,

...)l k .

380

Then

38 7

fC;T

/' . . . 0

... 0

r!

I (cr( r - k)!r!llAo + .l;-k)21 so that

. ..

382

which , together with

k=l

verifies the former part of (3) in THEOREM. The manner similar to the above argument ,together with (2) of Lemma 1, yields the proof of the latter half of (3) of THEOREM.

References [l] S. Albeverio and J. Schafer, Abelian Chern-Siomons theory and link

invariants, J. Math. Phys. 36 (1995), 2157-2169. [2] S. Albeverio and A. Sengupta, A Mathematical Construction of the Non- Abelian Chern-Simons Functional Integral, Commun.Math.Phys.

186 (1997), 563-579. [3] I.Ya. Aref’eva, Non-abelian Stokes formula, Theor. Math. Phys. 43 (1980), 353-356. [4] S.Axelrod and I.M.Singer, Chern-Simons perturbation theory, in “Proc. XXth. DGM Conf.,” World Scientific, 1992, 1-36. [5] D. Bar-Natan and E. Witten, Perturbation expansion of Chern-Simons theory with non-compact Gauge group, Commun.Math.Phys. 141 (1991), 423-440. [6] J.M. Bismut, “Large deviations and the Malliavin calculus,” Progress in Mathematics 45 Birkhauser Boston Inc., Boston, MA, 1984. [7] J.D.Dollard and C.N.Friedman, “Product integration,” in “Encyclopedia of Mathematics and its applications,” Vol 10, Addison-Wesley, Massachusetts, 1979.

383

[8] I.M. Gelfand and N.Ya. Vilenkin, “Generalized functions,” 4, Academic Press, 1964. [9] P.B. Gilkey, ‘‘ Invariance Theory, The Heat Equation, And the AtiyahSinger Index Theorem,” Publish or Perish, Inc., 1984.

[lo] E. Guadagnini, M. Martellini and M. Mintchev, Wilson lines in ChernSimons theory and link invariants, Nucl. Phys. B330 (1990), 575-607. [ll]N. Ikeda and Y. Ochi, Central limit theorems and random currents,

Lect. Notes in Control and Inform. Sci.78 (1986), 195-205.

[12] H.H. Kuo, “Gaussian Measures in Banach Space,” Lect. Notes in Math. 463, Springer, Berlin Heidelberg New York, 1975. [13] H.B. Lawson,Jr and M.L. Michelsohn

,

“Spin Geometry,” Princeton

University Press, Prinston, New Jersey,1989. [14] R. Leandre, Invariant Sobolev calculus on the free loop space, Acta Appl. Math. 46 (1997), 267-350. [15] P. Malliavin and S. Taniguchi, Analytic functions, Cauchy formula and Stationary phase on a real abstract Wiener space, J. Funct. Anal. 143 (1997), 470-528. [16] I. Mitoma, One loop approximation of the Chern-Simons integral, Acta Appl. Math. 63 (2000), 253-273. [17] I. Mitoma, Wiener space approach to a perturbative Chern-Simons integral, Canad. Math. SOC.Conf. Proc. 29 (2000), 471-480. [18] C.B. Morrey, Jr., “Multiple integrals in the calculus of Variation,” Springer, Berlin Heidelberg New York, 1966. [19] Y. Ochi, Limit Theorems for a Class of Diffusion Processes, Stochastics

15 (1985), 251-269. [20] J.N. Tavares, Chen integrals, generalized loops and loop calculus, Internat. J . Modern Phys. A 9 (1994), 4511-4548.

On the stochastic transport equation

of convolution type Habib Ouerdiane

DQpartement de Mathkmatiques, FacultQdes Sciences de Tunis UniversitQ de Tunis El Manar. Campus Universitair 1060 Tunis, Tunisie

Josk Luis Silva

University of Madeira, CCM 9000-390 Funchal, Portugal

Abstract In this paper we study the stochastic transport equation of convolution type. For general initial condition and its co-

efficients we give an explicit solution which is a well defined generalized stochastic process in a suitable distribution space. Under certain assumptions on the coefficients we also write the obtained solution as a convergent series of integrals.

Keywords and phrases.

Generalized functions, convolution calculus,

stochastic transport equation, generalized stochastic process.

AMS Subject Classification. Primary 60H15; Secondary 35D05, 46F25, 46G20. 384

385

1. Introduction

The aim of this work is to study the solution of the following Cauchy problem corresponding to the stochastic equation modeling the transport of a substance which is dispersing in a moving medium

where X ( t ,z , w ) is the concentration of the substance at time t E [0, m) and at the point z E

R‘, T E N,;a2 > 0 is the dispersion coefficient (constant),

A (resp. V) is the Laplacian (resp. the gradient) with respect to the spatial variable z, w

,

= (q, . . . w d ) is the stochastic

vector variable in the tempered

Schwartz distribution space 5’; := S’(R, R d ) with the standard Gaussian measure, d E

N,* is the convolution product between generalized functions

(see Subsection 2.2 below), f is the initial concentration of the substance,

V = (Vl,. . . ,Vr) is the vector velocity of the medium, K is the relative leakage rate and g is the source rate of the substance. The Cauchy problem (1.1)was analyzed by many authors from different point of view, see e.g., [7], [14] and references therein for more details and historical remarks. Recently Ouerdiane et al. [12] obtained the solution of (1.1)in the particular case when V

=g =0

in terms of the convolution exponential as a well

defined generalized function in a suitable distribution space. In addition for the case when K is a positive family of generalized stochastic processes

386

the corresponding solution is given as a limit of integrals, see [13] for details and also [I],[2] for related topics. The starting point is the following Gelfand triple

F;(N‘) 3 L 2 ( ~ y) ’ , 2 Fe(N’), where

N‘ is the dual of

a complex nuclear Frkchet space

N,Q

is a Young

function, y is the usual Gaussian measure on M‘ which corresponds t o the real part of N’. The test function space .FQ(N’) is defined as the space of all holomorphic functions on N‘ with an exponential growth condition of order

6’. The generalized function space FL(JV’) represents the topological dual of

Fe(N’). In the following we will choose the nuclear space n/ = ( s d x w‘)~, the complexification of the real nuclear space

s d

x

R‘,which is adapted t o

our situation. We would like to stress that all differential operators involved in equation (1.1) are interpreted in the generalized sense. Using the Laplace transform L we may define the convolution of two generalized functions

a, 9 E FL(N’) as CP * 9 = L-yLCP. L9)

which allows us to introduce the convolution exponential of exp* CP as an element in FG(N’), where the Young function

denoted by

‘p = ( e e * ) *

and

denotes the polar function associated to 8, see e.g., [8]. For positive generalized stochastic process

CP

=

(CP(t))t>o there exists a

family of Radon measures p = (pt)t>o - (see e.g., [ll])on M’ which represents V such that the Fourier transform of p t , t

2 0 is given by

JM’

where ((., .)) denotes the duality between

FL(N’) and Fo(N’) and

sponds t o the extension of the inner product of L2(M’,y).

corre-

387

The paper is organized as follows: in Section 2 we review some of the terminology and theory necessary for the stochastic model and its calculus. In particular, we define the Laplace transform, the convolution product on the space of generalized functions and establish some of its properties, i.e., the characterization of generalized functions and convolution exponential. These are the contents of Subsections 2.1 and 2.2. In Section 3 we give a general scheme for solving the convolution type equations as e.g., the Cauchy problem (1.1). Finally in Section 4 we write the solution of (1.1) as a limit of integrals. 2. Preliminaries

2.1. Test and generalized functions spaces In this section we introduce the framework and tools which is suited for the applications which are intended in later sections. For a general account on the distributions spaces and convolution calculus presented here the interested reader is referred to [5], [3] and the references quoted there. We start with a separable real Hilbert 7-l space which we choose to be

7-l = L2(R,Rd) x R ', d, T E N with scalar product (., .) and norm I . I. More precisely, if (f,.)

=

( ( f ~.,. . ,fd),

( q., . . , z')) E 7-l, then

Let us consider the real nuclear triplet

M'

= S'(R,

Rd) x R '

ZI

7-l ZI S(R,Rd) x R ' =M.

(2.1)

The pairing (., .) between M' and M is given in terms of the scalar product in'H7i.e.7

((W,Z),(t,Y))

:= (W,~)LZ(IW,IWd)+(Z,Y)WI.,.(W,Z) E M ' a n d ( t , y ) E

M . Since M is a FrQchet nuclear space, then it can be represented as

n s,(Iw,R~) 00

M=

n=Q

n M,, 00

R' =

n=Q

388

where S,(R,Rd) x

; 1 + I . I&.,

R' is a Hilbert space with norm square given by I .

see e.g., [6] or [4]and references therein. We will consider the

complexification of the triple (2.1) and denote it by

where N = M +iM and 2 = 7-l +iX. On M' we have the standard Gaussian measure y given by Minlos's theorem via its characteristic functional: for every ( J , p ) E M

C&P)

=

s,,

exp(i((wlx),( J , P ) ) ) w ( w , X ) )

1

= eXP(-5(IEl2

+ IPI2>).

In order to solve the Cauchy problem (1.1) we need to introduce an appropriate space of generalized functions. We borrow this construction from [9].

R,

+ e2(t2)where B1,02 are two

Let 0 = ( & , 0 2 ) : : R

4

Young functions, i.e.,

Bi : R+ 4 R+ continuous convex strictly increasing

(tl,t2) H

el(t,)

function and Qi(t)

lim -= 03,

t+W

t

For every pair m = (rn1,mz)with space Fo,,(N-,),

& ( O ) = 0, ml,m2

i = 1,2.

E]O,oo[,we define the Banach

n E N by

where for each z = ( w , x ) we have e(mlzI-,) := 6$(rn1lwl-,)

+ &(maIxI).

Now we consider as test function space the space of entire functions on N' of (81, &)-exponential growth and minimal type

389

endowed with the projective limit topology. We would like to construct the triple of the complex Hilbert space L 2 ( M ’ , y ) by

.Fo(N’).To this end we

need another condition on the pair of Young functions (01, 02). Namely, Qa

(t)

lim -< 03,

t-+m

t2

i = 1,2.

This is enough to obtain the following Gelfand triple

where

.Fi(N’)is the topological dual of .Fo(N’)with

respect to L 2 ( M ’ , y )

endowed with the inductive limit topology. In applications it is very important to have the characterization of gen-

.FL(N’).First we define the Laplace transform of an element in .FL(N’). For every fixed element ( J , p ) E N the exponential function exp((E,p)) is a well defined element in .Fo(N’),see [5]. The Laplace transform L of a generalized function @ E 3L(N‘)is defined by eralized functions from

We are ready to state to characterization theorem, see e.g., [5] and [12].

Theorem 2.1. The Laplace transform is a topological isomorphism between

3L(N’)and the space Gp(N),where Gp(N)is defined by

and GO.,~(N,) is the space of entire functions o n Nn with the following 0-exponential growth condition

390

2.2. The Convolution Product

*

It is well known that in infinite dimensional complex analysis the convolution operator on a general function space 3 is defined as a continuous operator which commutes with the translation operator. Let us define the convolution between a generalized and a test function. Let @ E FL(N’) and

‘p

where

E

FO(N’) be given, then the convolution @ * ‘p is defined by

T-(~,.)

is the translation operator, i.e.,

.Fo(N’).The convolution product is given for any @ E FL(N’) in terms of the dual pairing as (@* ‘ p ) ( O , O ) =

It is not hard the see that and

‘p

E

@*‘pE

3O(N’).

We can generalize the above convolution product for generalized functions as follows. Let @, 9 E FL(N’)be given. Then @ * 9 is defined as

This definition of convolution product for generalized functions will be used on Section 3 in order to write the solution of the stochastic heat equation given in (1.1).We have the following equality, see [12], Proposition 3.3:

As a consequence of the above equality and the definition (2.6) we obtain that

which says that the Laplace transform maps the convolution product in

FL(N’)into the usual pointwise product in the algebra of functions GO.( N ) .

391

Therefore we may use Theorem 2.1 to define convolution product between two generalized functions as @ * 9 = Lc-1(L@L9).

Relation (2.7) allows us t o define the convolution exponential of a generalized function. In fact, for every @ E exp(L@) E

G,@*(N).Using

FA(N')we may easily check that

the inverse Laplace transform and the fact

that any Young function 9 verify the property ( 9 * ) * = 9 we obtain that

LC-l(Gee* ( N ) )= .F[e.*,.( W ) .Now we give the definition of the convolution exponential of @ E

FA(N'),denoted by exp* @

Notice that exp* @ is well defined element in .F[e8.) * ( N ' )and therefore the distribution exp* @ is given in terms of a convergent series 0 0 .

exp*

= 60

+ C n!1 n=l

where

is the convolution of @ with itself n times, @*' := 60 by conven-

tion with 60 denoting the Dirac distribution a t 0. The following property follows easily from (2.8) and (2.7): if @, 9 E

.FA(N')then

exp* @ * exp* 9 = exp*(@* 9).

(2.10)

3. Applications to the stochastic transport equation

A one parameter generalized stochastic process with values in F'(N') is a family of distributions { @ ( t )t, 2 0) to be continuous if the map t

H

c FL(N'). The

process @ ( t )is said

@ ( t )is continuous. In order t o introduce

generalized stochastic integrals, we use the characterization theorem for sequences of generalized functions, see [lo], Theorem 3. For a given continuous generalized stochastic process (X(t)),,o we define the stochastic

392

generalized process

1 t

Y ( t , x , w )=

X ( s , z , w ) d s E FA(N’)

by

The process Y ( t , z , w ) is differentiable and we have

Y(t,z,w) =

X ( t , x , w ) . The details of the proof can be seen in 1121, Proposition 4.11. The results established up to now may be applied to a wide class of SPDE’s of convolution type. The general procedure is the following: Step 1: Assume that the functions involved in the SPDE can be modelled as some convolution functional and all products involved are interpreted as

convolutions products. Step 2: Apply the Laplace transform L to the SPDE. This produce a deterministic differential equation (with usual products) with the unknown

t H X ( t ,p , t) function where (pi t) E N . S t e p 3: Solve this deterministic differential equation and then by the characterization Theorem 2.1 X ( p , t) is indeed the Laplace transform of an element

X(., .) E Fb(N’) for a suitable choice of ,B which then solves the

origina1 equation. Let us apply this scheme to solve the Cauchy problem in (1.1). We recall again this problem for the reader convenience. Let f be a given generalized function in

Fh(N’) and V, K

and g .FA(N‘)-valued continuous generalized

stochastic process. Consider the following stochastic differential equation with initial condition f

393

To solve this SPDE we apply the Laplace transform to (3.2) and obtain

The solution of (3.3) is given as (using the method of variations of constants)

Now the solution of the system (3.2) is given using (3.1), (2.10) and (2.8) and the characterization theorem, Theorem 2.1. We give it on the next proposition.

Theorem 3.1. The Cauchy problem (1.1) has an unique solution X ( t ) which is a generalized Fh(N')-valued stochastic process, where the Young function ,D is given by ,D = (ee*)*. Moreover, the solution X ( t ) is given explicitly by

X ( t ,w , z) = f(z,w ) * y0zt * exp*

where

^/ozt

aV EL,az,

(1 t

[divV(s,5 ,w )

+ K ( s ,z, w ) ] d s )

i s Gaussian measure o n R' with variance a2t and divV

=

394

4. The solution of the transport equation as limit of integrals In this section we will write the solution of the Cauchy problem (1.1) as a limit of convergent series of integrals. In general, if we suppose that

W = (W(t))t>o - is a positive generalized stochastic process (i.e., V t 2 0

((W(t), 9)) 2 0 for any ‘p E To(”) with ’p(x+iO) 2 0 ’v’x E M ) represented by the family of Radon measures (pt)t>o, then for any t

20

Moreover the measure pt verify the following integrability condition: for any t

2 0 there exists n E N and m > 0 with pt(M-,)

= 1 such that

exP(e(mlYl-n))dPt(Y) < m.

(4.1)

For each Radon measure p on M’ verifying (4.1) and all and u = (x,w) E N’ = M’

‘p

E

.F@(N’)

+ iM’ we have the following equality, see [13]

Lemma 3.2 ((.P*

001

P)*’P)(U) = ‘p(U)+C

/

-J

‘p(U+Yl+. ‘ .+Yn)dP(Yl). . .dP(Yn).

(M’)”

n=l

(44 This equality may be generalized when

‘p

is replaced by a generalized func-

tion Q, E F;(N’). In fact, we have 0 0 -

n

where for every n = 1 , 2 , . . . the distribution dp(yn) is defined for any

‘p E

.F@(N’) as

s(M,)n T?,,

Yn Q,

dp(y1) . . .

395

(@ -L A Y

* ‘p)(Yl+. . . + Yn)&(Y1). . . dP(Yn).

For the details of the proof see 1131 Lemma 3.6. Moreover if (W(s)),?o

c .FL(N’) be

a positive generalized stochastic

process represented by the family of measures ( p s ) s i ~then , for any

‘p

E

.Fe(N’)we have

and consequently

In fact equality (4.4) is nothing but the definition (3.1) with

‘p =

exp((6,p)). Therefore by a limit procedure we get the required result (4.4) for general test function ‘p E

.F,g(N‘).To prove equality (4.5) we proceed in

two steps: first we notice that for every s 2 0 W ( s )* W ( s )is represented by ps * p,. Then iterating this process we obtain

Then equality (4.5) is a consequence of (4.4) and (4.6). Combining (4.5) and (4.2) with ( W ( S ) ) , ~ cOFL(N’) a positive generalized stochastic process represented by the family of measures ( p , ) , ? ~ and any test function

‘p

E .Fe(N’),u E

N‘

we have

396

If instead of (4.2) we use (4.3) then the equality (4.7) reads as

for every generalized function 9 E

.FL(N').

We are now ready to write the solution (3.5) of the Cauchy problem (1.1) as a convergent series of integrals. We will apply (4.8) for a suitable choice of (W(S)),>O - and 9.

Theorem 4.1. Let V, K be such that (divV(s) + K(s)),>o - is a positive

generalized stochastic process represented by the family of Radon measures (p,) t is given

f ( t , s)ds] is the

zero-coupon bond with maturity T. It can be written as

(3.4)

From the latter equations we obtain that the discounted forward asset value curve ?(t, .) depends only on

V, and the discounted forward growth

rates curve c(t,.)l a fact which will be used in the following sections.

We will now try to find the connection between asset value, discounted forward growth rates and share price of a company at a specified time t . We assume from now on that all details concerning the asset value, profits and debts/liabilities of the company are publicly available on the market, i.e. that V, and { c ( t , s ) : s 2 t } are known a t time t by all market participants (homogeneous beliefs). Following the intuitive ideas introduced in the beginning of this chapter, we know that the share price S,

at time t should depend on the asset value V, at time t, and the discounted forward growth rates curve of the company {G(t,s) : s 2 t } at time t4.

We will first introduce a restricted model with finite time horizon. We suppose that we are situated a t time t and that the company has a planing 4We assume that t h e forward interest rates curve a t time t is known.

407

horizon of T * ,where t

< T* < 00

can depend on t (i.e. we assume that the

company does not plan any specific economic activity after T* and that all existing company debt will be settled before T * ) . This implies that the specific growth rate in the asset value G(t,T)is zero for T

2 T* or

equivalently the asset value of the company will grow a t the same rate as the risk free forward interest rate, i.e. g ( t , T ) = f ( t , T ) for T 2 T*. In this context it is natural t o assume that the (discounted) forward asset value of the company V ( t ,T ) is finite for all T

for all T

> t , which is equivalent

> t , i.e.

that

to the following assumption :

lT*

s(t,s)ds

< co

The fact that the forward growth rate curve g ( t , .) and the forward interest rate curve f ( t ,.) are equal on the interval [T*,+m) implies that after the forecasting time horizon T* the asset value of the company V ( t ,7') will grow like a bond, i.e the discounted forward asset value will not change after T*:

V ( t ,T ) = V ( t ,T * ) ,

(3.7)

for any T 2 T*. At time T* the owner(s) of the company can sell the whole assets of the company for the price V ( t ,T * ) ,hence the forward market price of this company is worth (at least) V ( t , T * ) .An arbitrage argument implies that for any financial product which at some time T* is expected to be worth V ( t , T * )and starting from that time will grow with the risk neutral interest rate, i.e. behave like a bond, one can obtain the

408

market price at time t by discounting, i.e. the market price of that financial product a t time t is given by ?(t,T*).This implies that the market/share price of the company a t time t is given by the expression

or equivalently that the discounted share price satisfies

Remark: Although we may find a T,,

E

-

(0,T * )such that V ( t ,T,,,)>

?(t,T*), ?(tlTmaZ)is not a good candidate for the fair share price of exp JTiazG(t, u)du]= ?(t, T * )implies that ) [ C(t,u)du < 0, i.e. that the company has to settle up debts on the time

the company, since V ( t ,T,,,

ST:,

interval [T,,,,

-

T*].Hence by selling all assets and settling up all remaining

debts of the company at time T,,,,

the owners neglect the profits of the

company on the time interval [T,,,,T*]

and therefore cannot obtain a

better price than the one he could obtain a t time T*. The fact that we are working with a finite prediction horizon T* implies the existence of an easy connection between share price, forward growth rates and asset value of a company. This is at the same time a drawback of the model, since T* depends on t (the prediction horizon should increase when t increases). We will now lift the restriction of a finite time horizon and generalize the model for the case where T* = 00. The assumption (3.6) naturally extends

409

to

The discounted forward asset value of the company ?(tl T) is therefore known for all T 2 t and given by V, exp

implies that V ( t ,co) := limT*,, V(tl co) = V, exp

lim

s)ds]

. Assumption (3.10)

?(tl T*) exists and

[ lT* T*+m

[JTc(tl

c(t,s ) d s ]

= V, exp

[lm S(t, s)ds]

. (3.11)

In the model with finite time horizon T*, we know that the share price is given by S,(T*)= V ( t ,T * ) .Hence by letting the time horizon T* going to

+m1 the price S'jT*) converges to Sjm)= V,exp

[stmc(t,s)ds] . Therefore

a natural way t o define the share price St in the model with unrestricted time horizon (i.e. T* = +co) is t o set St = V, exp

[st"c(t1

s)ds].

Intuitively] t o obtain the share price we have to discount the forward asset value of the company a t infinity V ( t ,00).

4. HJM-type dynamics of the discounted forward growth

rates and the connection to the BS model In order to build a market model where we can compute prices of derivatives] we have t o specify one possible stochastic dynamics for the discounted forward growth rates. We assume that for each fixed T with 0 5 t 5 T

< co

the discounted forward growth rate ?(t,T) satisfies the following SDE (already known from the HJM model)

with initial conditions given by c(0, .) = p(0,.), where Wt is a Wiener process under the historical probability

P whereas the drift a(tlT) and

410

the volatility o ( t , T )> 0 are measurable processes. In order for the share prices to be well defined we aSsume that A(u),C(u)and

#,

u 2 0 are

locally bounded stochastic processes, where A(u) and C ( u ) are define by

A(u) =

suma(u,

s)ds and C ( u ) = Jt"

o(u,s)ds > 0.

We now look for a condition for the market to be arbitrage free. A sufficient condition is the existence of a (risk neutral) probability measure

P*locally

equivalent to P such that the discounted market price

st ia a

martingale with respect to P*. The calculations are very similar to the calculations in the HJM model introduced in the second chapter and we obtain from (3.9):

This shows that

st satisfies:

{

dgt = gt A ( t ) + where dW,* = dWt

dt

+ gtC(t)dWt= gtC(t)dW,*,

+ [A(t)/C(t)+ $ C ( t ) ]d t , t 2 0.

We note that

(44

gt and

W,* are well defined since C ( t ) > 0 and C ( t ) ,A ( t ) / C ( t are ) locally bounded.

411

Moreover if we introduce the martingales

L and Z defined by

1 Zt = €(L), = exp(-Lt - - < L 2

(where

< L >t=

(4.4)

>t)

s," C ( ~ ) ~ dwes ) can make a Girsanov transform6 and

define P*locally equivalent to P by motion under P*. This implies that

1~

= 2,such that

W: is a Brownian

P*is a martingale measure and that

the model is arbitrage free. The discounted share price is therefore given by

which is a P*-martingale and corresponds to the discounted market price of a company in the Black-Scholes model with stochastic volatility process C(s). Hence for the special case that the dynamics of the discounted forward

growth rates is given by the HJM-type stochastic differential equation (4.1) we obtained the Black-Scholes model, establishing therefore a connection between the Heath-Jarrow-Morton and the Black-Scholes models. 5. Modelling discounted growth coefficients by L evy

processes

For practical use forward growth rate curves { g ( t ,T ) ,T 2 t } ,t E [0, +m) have the inconvenience of being parameterized by two (continuous) variables (t and T). We will introduce the concept of discounted growth coefficients of a company for the time interval

[ t l ,t z ] ,tl

5 t 2 , as seen from time t

5Lt and Zt are real martingales since A ( t ) / C ( t )+ i E ( t ) are assumed to be locally bounded. 6See e.g. [3]

412

and denote them by E t [ t l ,t 2 ] . The connection between the forward growth rate curve g ( t , .) and the growth coefficients Et[t1,t z ] is given by

We note that the discounted growth coefficients are additive, satisfying

To get closer t o the "real world" we will assume that reports concerning the activity and increasing/decreasing of the asset value of a company will be available on the market only a t discrete time points TO= 0

< TI < T2 < ...

( e.g. through quarterly reports or annual shareholder meetings). For simplicity7 we will take into account only the first (n-1) reports T I ,...,T,-l concerning the company and set T,

=

co. We define

Gf

to be the dis-

counted growth coefficient of the company for the time interval as seen from time t, i.e.

Gt

[Tk-,, T k ]

= E t [ T k - l , T k ] . We note that for any time

t 2 T k the discounted growth coefficient

e2 is already known a t time t and

given by

We can express the discounted share price gt of the company a t time t i 0 by the

Et

gt = Voexp

by substituting into equation (3.9):

]

-

]

Gt[O,m) = Voexp [k:l C G F = Voexp

[ I

[G'fATk]

. (5.3)

Using the same formula for t = 0 we get: 7Assuming T, = cm for some n avoids the necessity of using an infinite factor model.

413

where we set Lf = 2;- 5;. The latter equation shows that the dynamics of the process completely given by the dynamics of the n-dimensional stochastic processes

(Lt)t,O and the initial value So. We also note that the dynamics of changes qualitatively after each

Tk

since Lf is stopped at

Tk

gt

and that the

asset value of the company V, (which is difficult to estimate in practice) does not appear explicitly in (5.4).

In the following we will describe a general setting where the discounted growth coefficients

(

, k = l..n are modelled by Levy processes, ( L i ,...,LT) is an adapted Levy processes on

9t€[0,Tk]

i.e. we assume that Lt =

(R, F,Ft,P)starting at 0 with characteristic triplets ( p , C, u). To obtain well defined prices of derivatives we have to make the additional assumption that Q t ( u ) = IE [e]is finite for t E [ O , c o ) and u E {uI,u2, ...,un}, where

at(.)

is given by the L evy-Khintchine type formula'

at(.)

= exp{t[
t- < u,C u >

(5.5)

In order to ensure that our market model is arbitrage free we look for a necessary and sufficient condition for the discounted market price

stto be

a martingale with respect to the pricing measure P. Using the independent 'See e . g . [ 5 ] .

414

-

increment property of the Levy process, St is a martingale if and only if = SOfor all tiO. For

t E [Tp,T p + l )we obtain the following equality:

A short iterative calculation shows that a necessary and sufficient condition for

st to be a martingale under IF'is given by @'1(Uk)

= 1,Vk = l . . n

which is equivalent to the following system of equations for k = l,...,n involving the characteristic triplet ( p ,0,v ) of (Lt) :

l

ccz,j+s,n n

e,Lj+z j=k

(e,z=kzi

n

-1-x

,Z,o of integral operators

coincides with the linearization of a strongly continuous semigroup with generator G. In [9], the families T, ans esG are called Chernoff equiva-

lent. The family T, is related to the conditional measure in a way that the Chernoff equivalence class determines the limit measure and that from the equation above, we may conclude convergence infinite dimensional distributions. The second part of the proof is then a tightness argument showing that we have indeed weak convergence. To obtain an idea how the effective potential is obtained, we compute the Chernoff equivalence class of the contraction family

where p M is the heat kernel on M , x E L and V L denotes the riemannian volume on L. To determine the Chernoff equivalence class of T,, we have thus t o understand the short time asymptotic of the intgral. By the Minakshisundaram-Pleijel formula ( [4] note that we consider the heat

449

equation

at - 1/2 A,

= 0)

we obtain (dim M

= m)

To actually compute this integral up to order

o(s),

we consider normal

coordinates E = ( E l , . . . , E l ) in L and normal coordinates X := in M , both centered around the point x E L embedding L

cM

c M.

(X’, ...,X m )

In these coordinates, the

will be denoted by X = F ( ( ) and we have the standard

expansion (see e.g. [6], p.8) detgL(J)

=

1

1 - gRicL,ij(0)EiEj

+ O(lE13)

for the determinant of the metric tensor. Since for s

-+

0 the mass is ex-

ponentially concentrated near x, we may substitute integration over L by integration over a normal coordinate neighborhood U, (z) for the computation of the polynomial short-time asymptotic. That implies

Since uL(d6) = J z ( J ) d J we obtain

... x 1 + -RicM,uv(0)Fu([)F”(E)

[

1’2

S + ;ScalM(O)

-

1

1 ;RicL,ij(0)t;i