Probability Theory and Harmonic Analysis (Pure & Applied Mathematics) 0824774736, 9780824774738

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PURE

AND

APPLIED

MATHEMATICS

A Series of Monographs and Textbooks

PROBABILITY THEORY AND HARMONIC ANALYSIS

Edited by J.-A. Chao WojborA. Woyczyhski

Probability Theory and Harmonic Analysis

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Earl J. Taft

Zuhair Nashed

Rutgers University New Brunswick, New Jersey

University of Delaware Newark, Delaware

CHAIRMEN OF THE EDITORIAL BOARD

S. Kobayashi

Edwin Hewitt

University of California, Berkeley Berkeley, California

University of Washington Seattle, Washington

EDITORIAL BOARD M. S. Baouendi Purdue University

Donald Passman University of Wisconsin

Jack K. Hale Brown University

Fred S. Roberts Rutgers University

Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas and University of Rochester Anil Nerode Cornell University

Gian-Carlo Rota Massachusetts Institute of Technology David Russell University of Wisconsin-Madison Jane Cronin Scanlon Rutgers University Walter Schempp Universitat Siegen

Mark Teply University of Wisconsin

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS

1. 2.

K. Yano, Integral Formulas in Riemannian Geometry (1 970) (out of print) S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) (out of print)

3.

V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, editor; A. Littlewood, translator) (1970) (out of print)

4.

B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation editor; K. Makowski, translator) (1971) L. Narici, E. Beckenstein, and G. Bachman, Functional Analysis and Valuation Theory (1971) D. S. Passman, Infinite Group Rings (1971)

5. 6. 7.

/,. Dornhoff Group Representation Theory (in two parts). Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971,1972)

8.

W. Boothby and G. L. Weiss (eds.), Symmetric Spaces: Short Courses Presented at Washington University (1972)

9.

Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, translator) (1972)

10.

L. E. Ward, Jr., Topology: An Outline for a First Course (1972) (out of

11. 12. 13. 14.

A. Babakhanian, Cohomological Methods in Group Theory (1972) R. Gilmer, Multiplicative Ideal Theory (1972) J. Yeh, Stochastic Processes and the Wiener Integral (1973) (out of print) J. Barros-Neto, Introduction to the Theory of Distributions (1973) (out of print)

15. 16.

R. Larsen, Functional Analysis: An Introduction (1973) (out of print) K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry (1973) (out of print)

17. 18. 19.

C. Procesi, Rings with Polynomial Identities (1973) R. Hermann, Geometry, Physics, and Systems (1 973) N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) (out of print)

20. 21. 22.

J. Dieudonne, Introduction to the Theory of Formal Groups (1973) I. Vaisman, Cohomology and Differential Forms (1973) B. -Y. Chen, Geometry of Submanifolds (1973)

23.

M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975)

24. 25.

R. Larsen, Banach Algebras: An Introduction (1973) R. O. Kujala and A. L. Vitter (eds.), Value Distribution Theory: Part A;

26. 27. 28.

Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973) K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) B. R. McDonald, Finite Rings with Identity (1974)

prin t)

29.

J. Satake, Linear Algebra (S. Koh, T. A. Akiba, and S. Ihara, translators) (1975)

30.

J. S. Golan, Localization of Noncommutative Rings (1975)

31. 32. 33. 34.

G. M. K. L.

35. 36. 37.

Topics (1976) N. J. Pullman, Matrix Theory and Its Applications (1976) B. R. McDonald, Geometric Algebra Over Local Rings (1976) C. W. Groetsch, Generalized Inverses of Linear Operators: Representation

38. 39. 40. 41. 42. 43.

and Approximation (1977) J. E. Kuczkowski and J. L. Gersting, Abstract Algebra: A First Look (1977) C. O. Christenson and W. L. Voxman, Aspects of Topology (1977) M. Nagata, Field Theory (1977) R. L. Long, Algebraic Number Theory (1977) W. F. Pfeffer, Integrals and Measures (1977) R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to

Klambauer, Mathematical Analysis (1975) K. Agoston, Algebraic Topology: A First Course (1976) R. Goodearl, Ring Theory: Nonsingular Rings and Modules (1 976) E. Mansfield, Linear Algebra with Geometric Applications: Selected

Real Analysis (1977) 44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) 45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M. Marcus, Introduction to Modern Algebra (1978) 48. E. C. Young, Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K. Segal, Topics in Group Rings (1978) 51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978) 54. L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals: An Introduction to Harmonic Analysis (1979) 54. J. Cronin, Differential Equations: Introduction and Quantitative Theory (1980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980) 56. /. Lawman, Foundations of Three-Dimensional Euclidean Geometry (1980) 57. H. I. Freedman, Deterministic Mathematical Models in Population Ecology (1980) 58. S. B. Chae, Lebesgue Integration (1980) 59. C. S. Rees, S. M. Shah, and C. V. Stanojevic, Theory and Applications of Fourier Analysis (1981) 60. L. Nachbin, Introduction to Functional Analysis: Banach Spaces and Differential Calculus (R. M. Aron, translator) (1981) 61. G. Orzech and M. Orzech, Plane Algebraic Curves: An Introduction 62. 63. 64. 65. 66.

Via Valuations (1981) R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) W. L. Voxman and R. H. Goetschel, Advanced Calculus: An Introduction to Modern Analysis (1981) L. J. Corwin and R. H. Szcarba, Multivariable Calculus (1982) V. /. Istratescu, Introduction to Linear Operator Theory (1981) R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces: A Comparative Study in Algebraic and Analytic Settings (1981)

67. 68. 69.

J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981) D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981) J. W. Brewer and M. K. Smith, eds., Emmy Noether; A Tribute to Her Life and Work (1981)

70.

K. H. Kim, Boolean Matrix Theory and Applications (1982)

71.

T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982)

72. 73.

D. B. Gauld, Differential Topology: An Introduction (1982) R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983)

74.

M. Carmeli, Statistical Theory and Random Matrices (1 983) J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of Topological Semigroups (1983)

75. 76.

87.

R. L. Faber, Differential Geometry and Relativity Theory: An Introduction (1983) S. Barnett, Polynomials and Linear Control Systems (1983) G. Karpilovsky, Commutative Group Algebras (1983) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The Commutative Theory (1983) I. Vaisman, A First Course in Differential Geometry (1984) G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1984) K. Hrbacek and T. Jech, Introduction to Set Theory, Second Edition, Revised and Expanded (1984) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The Noncommutative Theory (1984) B. R. McDonald, Linear Algebra Over Commutative Rings (1984)

88.

M. Namba, Geometry of Projective Algebraic Curves (1984)

89.

91.

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1 985) M. R. Bremner, R. V. Moody, and J. Patera, Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) A. E. Fekete, Real Linear Algebra (1985)

92. 93.

S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) A. J. Jerri, Introduction to Integral Equations with Applications (1985)

94.

G. Karpilovsky, Projective Representations of Finite Groups (1985)

77. 78. 79. 80. 81 . 82. 83. 84. 85. 86.

90.

95.

L. Narici and E. Beckenstein, Topological Vector Spaces (1985)

96. 97.

J. Weeks, The Shape of Space: How to Visualize Surfaces and ThreeDimensional Manifolds (1985) P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research

98.

(1985) J.-A. Chao and W. A. Woyczyhski, eds.. Probability Theory and Harmonic

Analysis (1 986) 99. G. D. Crown, M. H. Fenrick, and R. J. Valenza, Abstract Algebra (1986) 100. J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of Topological Semigroups, Volume 2 (1986)

Other Volumes in Preparation

and Harmonic Analysis Edited, by

J.-A. CHAO Cleveland State University Cleveland, Ohio

WOjBOR A. WOYCZYNSKI Case Western Reserve University Cleveland, Ohio

MARCEL DEKKER, INC.

A-73

fil

T77

mu lUOO (o°[ '7

Library of Congress Cataloging-in-Publication Data Main entry under title: Probability theory and harmonic analysis. (Monographs and textbooks in pure and applied mathematics ; 98) Papers from the Mini-Conference on Probability and Harmonic Analysis held in Cleveland, Ohio, 1983. Includes index. 1. Probabilities--Congresses. 2. Harmonic analysis-Congresses. I. Chao, J.-A. (Jia-Arng) II. Woyczynski, W. A. (Wojbor Andrzej), [date]. III. Mini-Conference on Probability and Harmonic Analysis (1983 : Cleveland, Ohio) IV. Series: Monographs and textbooks in pure and applied mathematics ; v. 98. QA273. A1P77 1986 519.2 85-27427 ISBN 0-8247-7473-6

COPYRIGHT © 1986 by MARCEL DEKKER, INC.

ALL RIGHTS RESERVED

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, micro¬ filming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 10 987654321 PRINTED IN THE UNITED STATES OF AMERICA

Preface

Influenced by the ground-breaking work of H. Steinhaus, R. E. A. C. Paley, A. Zygmund, R. Salem, and J.-P. Kahane, the study of the interaction be¬ tween probability theory and harmonic analysis is one of the "success stories" of mathematics over the past two decades. This interaction has provided a major impetus for the development of both areas. This volume is a collection of fifteen papers in probability theory and harmonic analysis, some of which are of an expository and survey nature while others present original results not previously published. Topics covered in the volume range from martingales, stochastic integrals, and diffusion processes on manifolds, through random walks and harmonic functions on graphs, and random Fourier series, to invariant differen¬ tial and degenerate elliptic operators, and singular integral transforms. The papers were submitted to us by the lecturers at the Mini-Con¬ ference on Probability and Harmonic Analysis (Cleveland, May 10-12, 1983) and by speakers at other seminars of the Probability Consortium of the Western Reserve. We would like to express our appreciation to the authors for their contribution to the success of this volume. The Probability Consortium of the Western Reserve has been in existence for about five years, pooling resources in probability theory and related fields from Case Western Reserve University, Cleveland State Uni¬ versity, Kent State University, and other institutions in the area. The Con¬ sortium's organizing committee initially consisted of John Chao, Jim Hendricks, Ken Hochberg, Olaf Stackelberg, and Wojbor Woyczynski, and was recently joined by Alex de Acosta. We would like to acknowledge the cooperation of the staff of Marcel Dekker, Inc., in the preparation of this volume for publication; we thank them for their fine efforts. Steven Reazor was helpful in putting together the indexes. 7 ' J.-A. Chao W. A. Woyczynski

'

.

'

• /

Contents

Preface

iii

Contributors

vii

1. Vector-Valued Singular Integrals and the H'-BMO Duality Jean Bourgain

1

2. An Extension of a Classical Martingale Inequality Donald L. Burkholder

21

3. Conjugate Harmonic Functions on Trees J.-A. Chao and Mitchell H. Taibleson

31

4.

The Wiener Sausage and a Theorem of Spitzer in Riemannian Manifolds Isaac Chavel and Edgar A. Feldman

5. A Simple Proof of a Theorem by G. David and J. -L. Joumd on Singular Integral Operators Ronald R. Coifman and Yves F. Meyer 6. Reversible Diffusion Processes Richard Durrett 7. Intertwining Kernels and Invariant Differential Operators in Representation Theory John E. Gilbert, Ray A. Kunze, and Peter A. Tomas 8. The Mean Exit Time from a Tube in a Riemannian Manifold Alfred Gray, Leon Karp, and Mark A. Pinsky 9. Decoupling of Martingale Transforms and Stochastic Integrals for Processes with Independent Increments Stanislaw Kwapierf and Wojbor A. Woyczyrfski / /

45

61

67

91

113

139

v

Contents

vi

10. Brownian-Path-Preserving Functions on IR+ and Cauchy Distributions in IRn Gerard G. Letac 11. Infinitely Divisible Measures on the Space of Continuous Functions Induced by Random Fourier Series and Transforms

149

167

Michael B. Marcus 12. Regular Points for First Boundary Value Problem Associated with Degenerate Elliptic Operators Daniel W. Stroock and Setsuo Taniguchi

183

13. General Fourier Sums with Vector Coefficients and Analogs of A(p)-Sets Stanislaw J. Szarek

195

14. Spaces Generated by Blocks Mitchell H. Taibleson and Guido Weiss

209

15. Harmonic Analysis on Schur Algebras and Its Applications in the Theory of Probability Lajos Takacs

227

Author Index

285

Subject Index

289

' /

Contributors

JEAN BOURGAIN

Vrije Universiteit Brussels, Brussels, Belgium

DONALD L. BURKHOLDER University of Illinois, Urbana, Illinois J.-A. CHAO* Washington University in St. Louis, St. Louis, Missouri ISAAC CHAVEL The City College of the City University of New York, New York, New York RONALD R. COIFMAN Yale University, New Haven, Connecticut RICHARD DURRETTt

University of California at Los Angeles, Los Angeles,

California EDGAR A. FELDMAN Graduate Center of the City University of New York, New York JOHN E. GILBERT University of Texas at Austin, Austin, Texas ALFRED GRAY University of Maryland, College Park, Maryland LEON KARP^

University of Michigan, Ann Arbor, Michigan

RAYA. KUNZE University of Georgia, Athens, Georgia STANISLAW KWAPIEN# Case Western Reserve University, Cleveland, Ohio

Current affiliations: ♦Cleveland State University, Cleveland, Ohio tCornell University, Ithaca, New York ^Herbert H. Lehman College of the City University of New York, Bronx, New York ^Warsaw University, Warsaw, Poland

vii

viii

Contributors

GERARD G. LETAC University Paul-Sabatier, Toulouse, France MICHAEL B. MARCUS Texas A&M University, College Station, Texas YVES F. MEYER Ecole Polytechnique, Palaiseau, France MARK A. PINSKY Northwestern University, Evanston, Illinois DANIEL W. STROOCK* University of Colorado, Boulder, Colorado STANISLAW J. SZAREK Case Western Reserve University, Cleveland, Ohio MITCHELL H. TAIBLESON Washington University in St. Louis, St. Louis, Missouri LAJOS TAKACS Case Western Reserve University, Cleveland, Ohio SETSUO TANIGUCHIt University of Colorado, Boulder, Colorado PETER A. TOMAS University of Texas at Austin, Austin, Texas GUIDO WEISS Washington University in St. Louis, St. Louis, Missouri WOJBOR A. WOYCZYNSKI Case Western Reserve University, Cleveland, Ohio

Current affiliations: ^Massachusetts Institute of Technology, Cambridge, Massachusetts tKyushu University, Fukuoka, Japan

Probability Theory and Harmonic Analysis

1

Vector-Valued Singular Integrals and the rf-BMO Duality JEAN BOURGAIN Vrije Universiteit Brussels Brussels, Belgium

1. INTRODUCTION During the last few years, a new class of Banach spaces has been introduced and studied (see [3-5]) for which the analogs of several classical theorems on martingales and singular integrals are true. The main part of this chapter is a sketch of some very recent results in this area obtained by the author. DEFINITION A real (or complex) Banach space X is called ^-convex pro¬ vided that there exists a positive symmetric biconvex function £(x,y) on X x X satisfying f(x,y)0. Let # denote the usual Hilbert transform on the circle T, i.e., lim f *Q r— 1 r
x a £-convex space. Thus THEOREM 3 For 1 < p < °°, there is a constant Cp(X) [related to £(0,0) again] such that for f 2, ll < C(p)llfllHlllgllBMO p which means that the BMOp-norm dominates the H-^-dual norm for p > 2. We now verify that, conversely, the X*-valued BMOp-norms are dom¬ inated by the H^-dual norm. Consider an Ej-atom I and use Holder’s inequality for 2 < p < °° to estimate {(1/| II) / | g - E.gl ^ l/p by III _1//p| (g, h) | , where h e LP , A

J

x

II h IIpt < 1, h supported by I and of mean zero. Estimate, again using Doob’s inequality. III

l/pll h II h1 j 1/P/T Ihl + III JI

1//p / max E [|E h | ] JI , k-1 k k>]

< III _l/p|II l/p||h II , + III _l/p| II l/p|| | h| *11 , < C(p) — P p1 — This implies the equivalence of the BMOp norms for 0 < p < °° and com¬ pletes the proof of Theorem 12.

5. THE HARDY SPACE H^-(T) Unless we specify otherwise, X will again be an arbitrary complex Banach space. Define HX =

Hx(T) =

G Lx(T);

= 0 for n < °}

To prove Theorem 6, we establish the surprisingly general PROPOSITION 13 There is a numerical constant c such that

J

sup |f(rei0)| = 0< r< 1

J

sup |f*P I < cIIfII 0< r< 1 r

/

/

-

Bourgain

14

holds for f e H^(T). The similar property holds if we replace the radial maximal function by the maximal function on the Stolz domain. Use the pointwise inequality If * P t < (Ifl2 * P )2 r ~ r

(25)

if f G H^-(T). From (25), Proposition 13 easily follows by applying the max¬ imal inequality for scalar L2 -functions. The proof of (25) reduces to the scalar case and factorization of H1-functions. The following corollary to Proposition 13 is again deduced by a trans¬ ference argument. LEMMA 14

Let f be an X-valued function on the infinite forms T of the i0k form f = 2' gk( 0i, . . ., 0k_!)e • Then, with Ek denoting the expectations with respect to the product cr-algebras, the inequality I sup IE [f ] | I k k

(26)

< cllfll

holds. Proof: Assume (by approximation) that the gk are X-valued functions with a finite spectrum. Define a function on T by translating 0k — 0k + n^ip, where {nk} is a sufficiently rapidly increasing sequence of integers so that, in particular, this function is in HX(T). It is clearly possible to proceed such are obtained by convolution in ip with Poisson

that the expectations 2.js

a

(i) For 1 < p < °°, the following inequality holds:

e (g> e II rs r s C

p

< Cp2 (p - l)-1 II -

Yars e r (g> e sC II

p

(the M. Riesz theorem for Cp). (ii) More generally, fixing an integer te Z, II

Y

2-' r-s>t

a

e (g) e II rs r s C

Y

p

< Cp2 (p - 1)-11| a e (g) e II ^rsr sC

(boundedness of triangular truncations on Cp).

p

Bourgain

18

Proposition 18 and Corollary 19 (ii) give an analog of the LittlewoodPaley result for LP(T): COROLLARY 20 For A = (ars) in Cp,(l < p < °°), define A„ = T. a e (g> e 0 “ rr r r A

=

T.

u , „k-l „k 2 < r-s< 2

a

e (8> e rsr s

a

Z

ifk>0

e (8) e rs r s

ifk 1

It is elementary to check that for (x,y) e IR2, 0 < u(x,y) - | x + y| 0, then u(x + h, y + k) > u(x,y) + cp(x,y)h + ^(x,y)k where 0(x,y) =y and tp(x,y) = x if |x| V |y| < 1 and 0(x,y) = >p(x,y) = sgn (x + y) elsewhere in IR2. By (7), U + «Z„-1> If n = 1, this is the inequality u(Zj) > u(Z0). If n > 2, the expression ^(zn-l)(-^n _ Xn_i) can be written as the product of dn and a bounded meas¬ urable function of vp .. ., vn, dp .. ., dn_1. Thus, since v is predictable, E(Z

J(X - X 4 = 0 n-1 n n-1

with a similar result for ip. Therefore,

,

/

' '

Burkholder

24

Eu(Z^) > Eu(Znl) >

•• > Eu(Zq) = u(0,0) - 1

Consequently, using (6), we have that P(lgnl > 1) = 1 - P(lbXn + aYnl < b - a) < 1 - P(l X | V — n

| Y I < 1) n

< E[u(Zn) - I(Zn)] where I is the indicator function of the set {(x,y) : I xI V IyI < l}- By (5), (8), and (9), u(Z ) - I(Z ) < IX + Y I n n — n n

= (b - a)If I n

Combining these inequalities, we obtain P(|g I > 1) < (b - a)llfnll1

(10)

It is a short step from (10) to the desired inequality for g*. Let t = inf {n : | gnl > l}. Let uk be the indicator function of the set {t >k}. Then (ux, u2,...) is predictable relative to f, n frAn = £ “kdk k=l n g = ). u v, d, 5TAn . k k k k=l and (gT/\n) is the transform of the martingale (fTAn) by v. Applying (10), we obtain

p(g*>i) = p,igTAni > i) < (b-.)inTAnii1 where g* = sup1 0 and £(x,y) < |x + y|

if |x| = 1 y I =1

(13)

This is a little simpler than the original condition of ^-convexity introduced in [4] but is equivalent to it as we now show- Recall that in the original con¬ dition £ is also symmetric and satisfies £(x,y) < Ix + yI

if |x| < 1 < Iy 1

(14)

First, (13) implies that £(x,y) < |x + y| if |x| V |y| =1. To see this, assume as we can that |x| < lyl =1 so that x + y ^ 0, and choose ol £ (0,1) so that z = o-1(x + y) - y satisfies I z| = 1. Then x = oiz - (1 - a)y and, by (13), £(x,y) < a£(z,y) + (1 - a)£(-y,y) < a\ z + y| = I x + yI Now let u(x,y) = £(x,y) V

|x + y|

= Ix + yl Then u is biconvex on B x B. vex on B; if lyl < 1, then u(* sphere of B. So suppose that aq £ (0,1). If |xx | < 1 < |x21 I x2 + y I, so that

if |x| V

|y| < 1

if |x| V

|y| > 1

For example, if lyl > 1, then u(* ,y) is con¬ ,y) is locally convex away from the unit lyl < 1 = |x| and x = c^Xj + a2x2 where the , then u(xx ,y) > |xx + y| and u(x2,y) =

u(x,y) = Ix + yl < Qfjlxj +y| + o2|x2 +y| < ajU^.y) + a2u(x2,y) and if |xx | , |x2| >1, the same inequality holds. Now let £0(x,y) = [u(x,y) + u(y,x)]/2. Clearly, £0 is symmetric and biconvex on B x B, £0(0,0) = £(0,0) > 0, and £0(x,y) < Ix + yl if |x| V |y| >1. Therefore, £0 satisfies (14).

Extension of a Classical Martingale Inequality

29

The condition of ^-convexity is related not only to the behavior of B valued martingale transforms but is also related to .the behavior of a large class of operators of interest in harmonic analysis. For example, the ^-con¬ vexity of B is a necessary and sufficient condition for the Hilbert transform to exist and be bounded on the Lebesgue-Bochner space L^(IR), 1 < p < «>. Sufficiency is due to McConnell and the author [6], necessity to Bourgain [2]. The proof rests in part on the equivalence, for a Banach space, of ^-convexity and the UMD property [4].

4. STOCHASTIC INTEGRALS Let M = (Mt)t>0 be a real right-continuous martingale with left limits, M*(w) = supt>(pMt(u))| , and IIMI^ = supt>0 II Mt lly. Then XP(M* > X) < HMIIj,

X > 0

(15)

a classical inequality that easily follows from its discrete version; again see Ville [11] and Doob [9]. Let V = (V^.)^>q be a predictable process [8] satisfying 0 < V^(co) < 1. The key result of-this section is that the M* appear¬ ing in the left-hand side of (15) can be replaced by N*, the maximal function of another right-continuous martingale N with left limits, if N is the stochas¬ tic integral of V with respect to M: N

=

j

V dM a. s.,

t > 0

[0,t] THEOREM 4 Let N be the stochastic integral of V with respect to M as above. If 0 < V < 1, then XP(N* > X) < IlMllj,

X>0

(16)

The discrete analog is Theorem 1, which implies (16) with the help of some of the work of Bichteler [1]; see [7] for the details in the parallel case

-i < V< i. Similarly, there is a stochastic-integral analog of Theorem 2: If a < 0 < b and a < V < b, the right-hand side of (16) is replaced by (b - a)IIM ||j.

REFERENCES 1. 2. 3. 4.

Bichteler, K. (1981), Stochastic integration and L^-theory of semi¬ martingales, Ann. Prob. 9, 49-89. Bourgain, J. (1983), Some remarks on Banach spaces in which martin¬ gale difference sequences are unconditional, Ark. Mat. 21, 163-168. Burkholder, D. L. (1966), Martingale transforms, Ann. Math. Statist. 37, 1494-1504. Burkholder, D. L. (1981), A geometrical characterization of Banach spaces in which martingale difference sequences are Unconditional, Ann. Prob. 9, 997-1011.

Burkholder

30

5. Burkholder, D. L. (1981), Martingale transforms and the geometry of Banach spaces, Lect. Notes Math- 860, 35-50. 6. Burkholder, D. L. (1983), A geometric condition that implies the exist¬ ence of certain singular integrals of Banach-space-valued functions, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, W. Beckner, A. P. Calderon, R. Fefferman, and P. W. Jones (Eds.), Wadsworth, Belmont, Calif. 7. Burkholder, D. L. (1984), Boundary value problems and sharp inequali¬ ties for martingale transforms, Ann. Prob- 12, 647-702. 8. Dellacherie, C., and P.-A. Meyer (1980), Probability et potentiel: theorie des martingales, Hermann, Paris. 9. Doob, J. L- (1940), Regularity properties of certain families of chance variables, Trans. Amer. Math. Soc. 47, 455-486. 10. Doob, J. L. (1953), Stochastic Processes, Wiley, New York. 11. Ville, J. (1939), Etude critique de la notion de collectif, GauthierVillars, Paris.

3 Conjugate Harmonic Functions on Trees J.-A. CHAO* and

MITCHELL H. TAIBLESON

Washington University in St. Louis St. Louis, Missouri

1. INTRODUCTION Properties of harmonic functions on a homogeneous tree have been studied by Cartier [3]; in particular, Fatou's theorem on convergence was estab¬ lished. In this chapter we introduce a version of conjugate harmonic func¬ tions and obtain characterizations of the Hardy space H1 by means of conju¬ gate transforms. Since the early 1970s, harmonic analysis on trees related to represen¬ tations of certain matrix groups over p-adics has been investigated by Serre [18], Cartier [3,4], and others. More recently, Figa-Talamanca and Picardello [14] and Mantero and Zappa [17] treated some aspects of harmonic analysis on trees related to harmonic analysis on free groups. See [14] for details as well as references. This chapter is closely related to results of Koranyi et al. [16], where the relationship between random walk martingales and boundary martingales associated with harmonic functions on a tree is examined. We state some preliminary results in Section 2. On a homogeneous tree of order q + 1 with q > 3, the notion of conjugate harmonic functions is comparatively easy to study. We treat the case q = 3 in some detail in Section 3. The case q = 2 is more subtle and is considered in Section 4. Some of the ideas used there can be applied to harmonic functions on some "trees" with "local loops."

2. PRELIMINARY RESULTS We consider a homogeneous tree of order q + 1 (q > 2), that is, an infinite connected graph with no nontrivial closed loops in which every vertex has *Current affiliation:

Cleveland State University, Cleveland, Ohio 31

Chao and Taibleson

32

exactly q + 1 edges. Let X = {$} be the set of all vertices. Given two vertices £ and £', there is a unique chain joining £ to £', i-e -, a sequence {t t ... t } of distinct vertices where |0 = £ and = £'• We set d(^ |') = m. We consider the so-called "compact" case. Call a fixed vertex goj the center. The boundary 9X of the tree is the collection of infinite se¬ quences to = («! ,uo ,w_i. • • •) such that d(“j>“j-l) = 1 and j = i, o, -l, .... For go = (wj ,w0 ,w_i, (u,j) = {x =

(Xj ,Xo

• • •) e

^ «j+i*

9X,

,x_! ,...): x. = ., j < i < 1}.

3 = 1> °> _1»

Identify these (w,j) as open sets in 9X. They form a basis for a topology in 9X. A metric on 9X is given as follows: If x = y, then 6(x,y) = 0. If x + y, then there is a smallest j such that Xj = yjThis implies that (x,k) = (y,k) for all j < k < 1. Let fqj d(x,y) = | q + i

if 3 < 0 ifj = 1

Hence X and its boundary are split into q + 1 branches and k) Up < 00 • i

Denote by the cr-field generated by {(x^, k) : £ = 1, . •., (q + l)q j, k < 0. Such a martingale f(x,k) on X is just a regular martingale (in partic¬ ular, a q-martingale) relative to {&\Jk1,

k < 0

when |x| < 1,

k < 0

P(x,k) = 2

(2

k

V 1x1)

2

and

Q(x,k) =

0

for k > -1

0

when | x| > 1 or | x| < 2 , k+1.

-2i(x)(nr!p')

when 2

< |x| < 1* k'< -2

k < -2

42

Chao and Taibleson

where it is the multiplicative character we referred to earlier, which takes values ± 1 •

ACKNOWLEDGMENT The research of the authors was supported in part by National Science Foun¬ dation Grants DMS-8200884 and DMS-8443708.

REFERENCES 1. Burkholder, D. L- (1966), Martingale transforms, Ann. Math- Statist. 37, 1494-1504. 2. Burkholder, D. L., and R. F. Gundy (1970), Extrapolation and inter¬ polation of quasi-linear operators on martingales, Acta Math. 124, 249-304. 3. Cartier, P. (1972), Fonctions harmoniques sur un arbre, Symp. Math. 9, 419-424. 4. Cartier, P. (1973), Geometrie et analyse sur les arbres, Lect. Notes Math. 317, 123-140. 5. Chao, J. -A. (1974), spaces of conjugate systems on local fields, Studia Math. 49, 267-287. 6. Chao, J.-A. (1975), Maximal singular integral transforms on local fields, Proc. Amer. Math. Soc. 50, 297-302. 7. Chao, J.-A. (1975), Lusin area functions on local fields, Pacific J. Math. 59, 383-390. 8. Chao, J.-A. (1979), Conjugate characterizations of H1 dyadic martin¬ gales, Math. Ann. 240, 63-67. 9. Chao, J.-A. (1982), Hardy spaces on regular martingales, Lect. Notes Math. 939, 18-28. 10. Chao, J.-A., and S. Janson (1981), A note on H1 q-martingales, Pacific J. Math. 97, 307-317. 11. Chao, J.-A., and M. H. Taibleson (1973), A subregularity inequality for conjugate systems on local fields, Studia Math. 46, 249-257. 12. Chao, J.-A., and M. H. Taibleson (1979), Generalized conjugate sys¬ tems on local fields, Studia Math. 64, 213-225. 13. Coifman, R. R., and G. Weiss (1977), Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83, 569-645. 14. Figa-Talamanca, A., and M. A. Picardello (1983), Harmonic Analysis on Free Groups, Marcel Dekker, New York. 15. Janson, S. (1977), Characterizations of H1 by singular integral trans¬ forms on martingales and IRn, Math. Sc and. 41, 140-152. 16. Koranyi, A., M. A. Picardello, and M. H. Taibleson (1985), Random walk martingales and boundary martingales associated with harmonic functions on trees, Proc. Conf. Harmonic Anal., Cortona, Italy, July 1984. To appear.

Conjugate Harmonic Functions on Trees

43

17. Mantero, A. M., and A- Zappa (1983), The Poisson transform and representations of a free group, J. Funct. Anal. 51, 372-399. 18. Serre, J.-P. (1977), Arbres, amalgames et SL2, Asterisque 46. 19. Taibleson, M. H. (1975), Fourier Analysis on Local Fields, Princeton University Press, Princeton, N.J.

**

•

'

'

' /

4 The Wiener Sausage and a Theorem of Spitzer in Riemannian Manifolds ISAAC CHAVEL

EDGAR A. FELDMAN

The City College of the City University of New York New York, New York

Graduate Center of the City University of New York New York, New York

1.

INTRODUCTION

In this chapter we discuss some results pertaining to the radial and time asymptotics of the Wiener sausage in Riemannian manifolds. Our discussion starts with the radial asymptotics for hyperbolic 3-space (actually, this is preceded by some background in the first two sections), where we sharpen an otherwise universal result (from Chavel and Feldman [6]). Then we con¬ sider the time asymptotics of the Wiener sausage in a Riemannian symmet¬ ric space of nonpositive curvature, where we may employ an asymptotic formula which is originally due to Spitzer [25]. Finally, we comment on some of the details of Spitzer's argument.

2.

PRELIMINARIES OO

We let M be a connected n-dimensional, n > 1, C Riemannian manifold. If x : U — lRn (where IRn denotes n-dimensional Euclidean space) is a coordi¬ nate chart on M, then the Riemannian metric on M, when restricted to U, is given, relative to the natural coordinate vector fields

a.

J

ax3

j = 1, • • •, n, by the matrix (gjk),

j, k = 1, ..., n. It is convenient to write 9 = (gjk).

s?"1 = (gjk)

g - det s

45

Chavel and Feldman

46

The Riemannian measure on M, dV, is given on U, relative to this chart,

by dV = \fg dx1 • • • dx° Associated to the Riemannian metric is the Laplace-Beltrami operator A, acting on C2 functions on M, and given by

a = -jz Tj 3.(^i

)

The Laplace-Beltrami operator determines a heat kernel p(x,y,t) for the heat operator

acting on functions on M x (O.+^o). The examples we have in mind are: (a) M is compact; (b) M is noncom¬ pact, but has smooth boundary and compact closure, in which case the heat kernel is the Neumann heat kernel (thereby describing heat diffusion in a bounded regular domain with insulated boundary); and (c) M is arbitrary noncompact, in which case the heat kernel is the minimal positive heat ker¬ nel. In the first two cases the heat kernel is unique and satisfies the conser¬ vation of heat property

J P(x,y,t) dV(y)

= 1

M for all x 0. In the last case the heat kernel is unique, by defi¬ nition, but it need not satisfy the conservation property (especially if M has smooth boundary and compact closure, in which case p is the Dirichlet heat kernel). The first two heat kernels are related to the Sturm-Liouville decom¬ position of the Laplace-Beltrami operator as follows. Let {o = Aj < A2 < ••• t+°°} denote the spectrum of M, with each eigenvalue repeated according to its multiplicity; and let { 00 > 01 > 02 * • • •} be a complete orthonormal sequence in L2(M), with each 0. an eigenfunction of Aj, i.e., 3 A0. + A.0. = 0 J J J Then p(x,y,t) has the expansion

Wiener Sausage in Riemannian Manifolds

47

' /

which implies that

/ p(x, x, t) dV(x) M

Ze

(1)

We introduce Brownian motion for each of our heat kernels. In case (a), where M is compact, the path space W consists of all continuous paths w : [o, +oo) — M. In case (b), where the heat diffusion is governed by the Neumann heat kernel, the path space consists of all continuous paths co : [0, + «> ) — M. In case (c), of the arbitrary noncompact manifold with minimal positive heat kernel, first form the one-point compactification M* = :MU{«} of M, and let the path space consist of those continuous paths co : [0, +°° ) — M* such that co(0) is in M and such that if co(t0) = °° for some t0 > 0, then co(t) = °° for all t >_t0. For any co in #we let £( 0 : co(t) = « } To each x in M we associate a probability measure Px on W, supported on those paths starting at x and determined by Px(^(t) £ E) = / p(x,y,t) dV(y) M where 3C is the random Brownian path in W and E is any Borel set in M. Note that in case (c) we have PY(£ > t) = / p(x,y,t) dV(y) X M The integral of a function on J?t with respect to the measure Px is denoted by Ex(f); and the integral over a subset gg in W' is denoted by Ex(f ;a). Given any Borel set E in M, we let TE(#) denote the first hitting time of E by iT, i.e., T^(5T) = inf {t > 0 : J'(t) e E}

3.

HITTING TIMES FOR SMALL GEODESIC DISKS

Given Xq in M and £ > 0, we let B(Xo ;e) denote the geodesic disk in M cen¬ tered at x0 and having radius equal to e. Then, in all our cases (a), (b), and (c), we have for n > 2 (2) Of course, the formula is false for n = 1. /

> ••

48

Chavel and Feldman

A sharper form of (2) is the explicit formula (Chavel and Feldman [6])

Px(TB(*o;e) 1

t]

~

(n " 2)Vl£n'2 / P dr

(3)

as £ i 0, where n > 2 and cn_j denotes the (n - 1) area of the standard unit sphere Sn_1 in lRn; and t Px(TB(Xo;e) * l) ~ mIt| /

dT

(4)

as e | 0, when n = 2. An immediate application of (2) is that if q£ denotes the heat kernel of f2£ = : M\ B(Xo ; e) vanishing whenever one of the space variables approaches S(x0;e), the com¬ mon boundary of B(Xo ;e) and £ , and otherwise, as prior to the deletion of B(x0;e), then (5) on (M\{Xq}) x (M\{Xq}) x (0, + «>). If {0 < A-i (e) < A.2(£) dV(y)

Then (3) and (8) imply (7) on the average Px- To obtain (7), in probability, we then study Ex(Vt e) using (8). The details are carried out in Chavel and Feldman [6]. Application to the Lenz shift is discussed in the references above. Here we are interested in the validity of (7), almost surely Px, for all x in M. We consider below the case of the hyperbolic 3-space IH3 since the special functions associated to the problem are exceptionally simple. The same comment holds for the 3-sphere SB3. For 1R3, or more generally, IRn, n > 3, the result is known (see Kac [13], Simon [24, p. 239], and Spitzer [25]). It seems quite reasonable that we should be able to extend the arguments below to all simply connected, rank 1, symmetric spaces. At this point, the argument seems rather messy, so we stick with IH3. First, recall the explicit Laplace transforms r -At-3/2 -a/4t TjT -\l a A Jet e dt = /—e a 0

(9)

and

I 0

t At ,, r~ -3/2 -{r+a/4r} , \f47re Q dT = e _At dt r A \fa 0

f

(10)

Next, recall that the heat kernel on IH3 is given by

P(x,y,t) =

e

-t -d2(x,y)/4t e d(x,y) n/2 (47rt) sinh d(x,y)

To calculate explicitly U(X,t) =J Px(TB(y; £) i t) on IHn, for all n > 2, first note that the spherical symmetry of !Hn implies that u depends only on d(x,y) and t. Next, pick geodesic spherical coordinates about y. Then the heat equation for u reads as /

1W 9u 9u (n - l)(coth r)V — = — v ' 9r at

with r = d(x,y) and u satisfying the initial boundary data u(r,0) = 0

for all r > e

u(e,t) = 1

for all t > 0

Of course, r(r,t) = 1 for all r < e, t > 0. If v is the Laplace transform of u, i.e.,

Wiener Sausage in Riemannian Manifolds

51

_^ e

u(r,t) dt

0 then 9zv 9v 8iT + (n ~ l)(coth r) ^ = Av

v(e,A) = which implies that

,

Vsinhe',n/2-1^ > , The crude version is

'W 12 ll MM E*dv

< const ez{t + c\Tt + e2}2 [the first inequality coming from the strong Markov law, and the second from (12)], i.e., e_zE (V2 X

) < const {t + c\ft + e2}2

tj £

For any fixed integer N > 1 we have N£"2Ex(Vt2/N e) — const N{N_1 + eN 2 + e2}2 For the moment, fix t, £ > 0, and a positive integer N. For each j = 0, ..., N - 1, let Vj denote the volume of tubular neighborhood, of %([tj/N, t(j + 1)/N]), having radius e. Then N-l

!Ex | = |//Q1>t(g){(LtPt)f-Lt(l)Pt(f)}^p

< (//lQ1 t(e)l2^)2[//l /it(x,y)(Pt(f)(y) - Pt(f)(x))/dy)2—-]

Coifman and Meyer

64

A simple application of Plancherel's theorem permits us to estimate the first term. For the second we use (1) and Minkowski's inequality to dominate it by

JJJ Pt(* - y) I Pt(f)(y) - Pt(f)(x)i2 dx-^-'J] =

[IIIPt(u)I Pt(f)(y) - Pt(f)(y + u) 12

2

= [///Pt(u)!^)l2le^-l|2|f(^^^]

///pt(-)|rl i^)i2itn6if(i)i2^f^]

= [/p(u)u6du] [/ 0

I0(t)l2t6^]

[/lf(|)l2d^]

As for the first term, we proceed as above and are led to estimate

* b(x) |2 | 0t * f|2—] This is dominated by llfll2 if and only if b is in BMO. To see how the theorem can be applied to study

we check by a simple integration by parts that «’n(a, l)(x) = ®n_1(a, a)(x). If we make the induction hypothesis that g'n_1(a,f) is bounded on L2 it would follow by standard methods that «’n_1(a,f) maps L°° to BMO. The verifica¬ tion of (1) is also fairly easy for this example (exploiting the antisymmetry °f ®n) • We conclude with a few observations concerning the main term 00

J

r

Ht

QtWt*b)(t*f))-

This bilinear operation on BMO x LP is the so-called paraproduct and has also been used by Bony [1] in his study of propagation of singularities of nonlinear partial differential equations. It can be rewritten in a simpler, essentially equivalent form as

Proof of a Theorem by David and Journe

65

permitting us to describe the corresponding martingale version (as a stochas¬ tic integral) as oo

'

' /

*(M) - £ vwv where f^ is an L? martingale and bk is a BMO martingale (see [2]).

REFERENCES 1.

2. 3.

Bony, J. M. (1981), Calcul symbolique et propagation des singularites pour les Equations aux derivees partielles non lin^aires, Ann. Scient. Ecole Norm. Sup., 4eme s£r. 14, 209-246. Coifman, R. R., and Y. Meyer (1978), Au-dela des opd’rateurs pseudodifMrentiels, Asterisque 57. David, G., and J.-L. Journ£ (1985), A boundedness criterion for gen¬ eralized Calderdn-Zygmund operators, Ann, of Math. 120, 371-397.

' /

6 Reversible Diffusion Processes RICHARD DURRETT* University of California at Los Angeles Los Angeles, California

1.

INTRODUCTION

This chapter is based on a talk given at a meeting of the Probability Consor¬ tium of the Western Reserve, May 10-11, 1984, and on a number of lectures given at UCLA during the 1983-1984 academic year. The main purpose of those talks and of this review is to give an exposition of some results of Ichihara (1978) concerning the recurrence and transience of reversible dif¬ fusion processes and to apply these theorems to obtain some information about the behavior of Brownian motions on manifolds and one new result concerning (reversible) diffusions with random coefficients (see Corollary B.4 in Section 6).

2.

BASIC DEFINITIONS AND OTHER PRELIMINARIES

Depending on one's viewpoint, the central object of study in this chapter is (1) an elliptic operator in divergence form or (2) a reversible diffusion process X^, t > 0. From the first viewpoint our central focus will be on elliptic operators which can be written as

where (i) (ii)

a, c

e

OO

C

aU(x) = a^(x)

*Current affiliation:

Cornell University, Ithaca, New YoYk 67

Durrett

68

(iii) says that L is elliptic at each x e Rd and when combined with (i) that L is uniformly elliptic on compact sets. We assume (i) only because we are lazy. The results below are valid for coefficients that are much less smooth, but the attendant technicalities would only serve to obscure the simple ideas on which the proofs are based, so we will keep to the modest level of generality in (i). It is by now well known that with each second-order elliptic differential operator of the type described above there is an associated (minimal) diffu¬ sion process X^, t > 0, and that this process can be constructed in a number of different ways (see Dynkin, 1962; Friedman, 1975; Stroock and Varadhan, 1979; or Ikeda and Watanabe, 1981), so we will just state the facts we will need about these processes and refer to the reader to the sources listed above for more details. A diffusion process X is a collection of processes xf, t > 0, x e Rd, with X^ = x which have continuous paths and satisfy: 1.

Let rn = inf{t : |X^| > n}. If f e C°°, then tA T f(XtAT > - / n 0

2.

n Lf(^ds

is a martingale. If t > tm = lim 7*n we set xf = A, where A is the point at 0

and

/^(l

OO

9

K< 0

(since 1 + Kr2/4) ~ cr as r — 2/\l -K), so M is a complete metric space when K < 0 but not when K > 0. When K > 0, M is the image of a punctured sphere under stereographic projection. When K = 0 it is the usual Euclidean space and when K > 0, it is "hyperbolic space." [For more details on this, see Spivak (1979, Vol. n, pp. 334-340).] Finally, we have a general class of examples that can be studied with¬ out a knowledge of Riemannian geometry. EXAMPLE 3c Manifolds with a "Pole" A point x e M in a complete Rieman¬ nian manifold is said to be a pole if the exponential map associated with the point x is a diffeomorphism from Mx (the tangent space at x) onto M. If the words in the last sentence do not make sense, don't worry. In the situation described there we have a nice global coordinate system defined on the mani¬ folds (so-called normal coordinates), so we can forget about M and suppose instead that we are in Rd with

Reversible Diffusion Processes

75

where x.

? A*> f^r =

x.

hi

For more about this class of manifolds, see Greene and Wu (1979).

4.

GENERAL RESULTS, COMPARISON THEOREMS

In this section we describe some general results concerning the recurrence and transience of reversible diffusion processes. The first order of business is to give some definitions. The process X is said to be recurrent if for each open set U and x e R^, Px(^t ^ U for some t > 0) = 1 and otherwise it is said to be transient. Deciding whether a given multidimensional diffusion process is recur¬ rent is a very difficult and largely unsolved problem, but in the reversible case the task is made considerably simpler by the "Dirichlet principle," which we will use in the following form: PROPOSITION 1 Let G be a bounded open set in Rn with smooth boundary. Let f be a continuous function on 8G and let Uf(x) = Ex(XT), where r = {t : X(- ^ G}. Then uf minimizes

within the class of functions which are C ous on G.

in G, are = f on 3G, and continu¬

For a statement and proof of the real theorem, see Ichihara (1978, pp. 447-449). It is essentially due to Littman et al. (1963). We will content ourselves here to make it plausible by sketching the proof. To begin we observe that V 9u ij 9v — + e>D(v) D(u + ev) = D(u) + 2e / 2 “ It ax. a dx i G ij J so if Uq is a minimum

and if v is a test function with compact support in G, integrating by parts gives

76

Durrett

0

=

J v Lug dx G

The last equality holds for any v, so we must have Luo = 0. It is easy to see that sup Ug < sup f and inf u0 > inf f, so UQ(Xf.AT) is a martingale and letting t-» we see that

vx> ■ vw - Extpv = vx>

"proving" Proposition 1. Given the last result it is clear what we should do to see if X is recur¬ rent. Let An = {x : 1 < |x| < n} = "the annulus of outer radius n and inner radius 1" and solve: Lu = 0

in An

u = 1

on {x : |x| = l}

u = 0

on {x : | xj = n}

The solution is u (x) S ) n Px 3, we have the following: COROLLARY la If d < 2 and a(x) < KI, where K is a constant < °°, then X is recurrent. COROLLARY lb If d > 3 and a(x) > el, where e is a constant > 0, then X is transient. By comparing with other processes we can get other results. For these purposes the following examples are useful. EXAMPLE 4 Models Suppose that a(x)x = f(|x| )x; i.e., the radial vector is an eigenvector of a with the eigenvalue depending only on Ix| . This occurs, for instance, if a(x) = f(|x|)1, then X is transient. These two corollaries contain results of Brown (1971, Corollaries 4.3.3 and 4.3.4) as a special case. By combining the last result with some standard comparison theorems in geometry, we can rephrase the results above in terms of the (1) Ricci and (2) sectional curvatures of manifolds (see Ichihara, 1982 for details).

Reversible Diffusion Processes

5.

79

SUFFICIENT CONDITION FOR RECURRENCE

As a consequence of Proposition 2 we see that if there is;a sequence of functions fn with fn = 1 on | x| =1 and fn = 0 on | x| = n which has D(fn) — 0, then since D(un) is the minimum D(un) < D(fn) — 0 and X is recurrent. Combining the last observation with a special sequence of test functions leads to the following result: THEOREM A Let

and let

J(r) = /d_1J(ro‘)d7r(°’) where n is a surface measure on Sd 1 normalized to be a probability measure. If OO

then X is recurrent. Proof: Given the hypothesis, it should not be too surprising that we let

f (x) = ip (I x|)

n

rn

By design ipn(n) = 0 and i^n(l) = 1, so fn satisfies the boundary conditions. Now Vfn = ^(|x|)(x/|x|), so

A

n

/

A

a«7fr)dx

n

1

,d-l S

/

Durrett

80

where | Sd = the surface area of Sd . Now ^>n(r) = In(r)/ln(l) = -l/r^wyi), so the last expression above

= IS

n

d-1 i /

1

= IS

d-1.

r

~2 dr J(r) d-1 J(r)In(l)

1

1

r 2 ^ In(l) 1

n

-.d-1,

J

dr d-1r J(r)

Xn(1)

which — 0 as n — 00 if (and only if) dr

y1) - f” 1

°°

as n — °°

rd 1J (r)

and in view of the remarks made above we have proved Theorem A. If we apply Theorem A to a model, then J(r) = f(r) and we have COROLLARY A. 1 In a model X is recurrent if oc

I

dr

OO

rd_1f(r)

so the result is sharp in case. In view of the proof this should not be sur¬ prising, but this helps explain the choice of In(r): it is a natural scale for the model. In Example 2 we have a1J(x) = e^(x) 3, .. a ^

j 0} Then X is transient. Proof: For a corresponding to directions in Va we have K(rcr) = 1 for all 0 < r < °° and hence oo

J

dr r1 ^K(rcr) < °°

1 (here we need d > 2), so the result follows from Theorem B. The proof given above makes the result obvious. To convince yourself that it is also incredible, observe that with probability 1 Brownian motion will exit Vq,, so if we drop the restriction that L be in the form (*), it is easy to see that we can define L on V° so that X is recurrent—but the details are messy to write down, so we will leave this as an exercise for the reader. In Example 2 we have a*j(x) = e0(x)