Stochastic Processes - Mathematics and Physics: Proceedings of the 1st BiBoS-Symposium held in Bielefeld, West Germany, September 10-15, 1984 (Lecture Notes in Mathematics, 1158) 3540159983, 9783540159988

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Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann

1158

Stochastic Processes Mathematics and Physics Proceedings of the 1st BiBoS-Symposium held in Bielefeld, West Germany, September 10-15, 1984

Edited by S. Albeverio, Ph. Blanchard and L. Streit

Spri nger­Verlag Berlin Heidelberg New York Tokyo

Editors Sergio A. Albeverio Mathematisches Institut, Ruhr-Universitat Bochum 4630 Bochum, Federal Republic of Germany Philippe Blanchard Ludwig Streit Fakultat fUr Physik, Ilniversitat Bielefeld 4800 Bielefeld, Federal Republic of Germany

Mathematics Subject Classification (1980): 03xx, 22xx, 28xx, 31xx, 34Bxx, 35xx, 35Jxx, 46xx, 58xx, 60Gxx, 60Hxx, 60Jxx, 60J45, 73xx, 76xx, 81 Fxx, 81 Gxx, 82xx, 85xx, 93xx ISBN 3-540-15998-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15998-3 Springer-Verlag New York Heidelberg Berlin Tokyo

Library of Congress Cataloging-in-Publication Data. BiBoS-Symposium (1st: 1984: Bielefeld, Germany) Stochastic processes, mathematics and physics. (Lecture notes in mathematics; 1158) Bibliography: p. 1. Stochastic processes-Congresses. 2. Mathematics-Congresses. 3. PhysicsCongresses. I. Albeverio, Sergio. II. Blanchard, Philippe. III. Streit, Ludwig, 1938-. IV. BielefeldBochum Research Center Stochastics. V. Series: Lecture notes in mathematics (Springer-Verlag); 1158. 0A3.L28 no. 1158 [0A274.A1] 510 s [519.2]85-26088 ISBN 0-387-15998-3 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

Preface The "1st BiBoS Symposium Stochastic processes: Mathematics and Physics" was held at the Center for Interdisciplinary Research, Bielefeld University, in September 1984. It is the first of a series of symposia organized by the Bielefeld - Bochum Research Center Stochastics (BiBoS), sponsored by the Volkswagen Stiftung. The aim of the topics chosen was to present different aspects of stochastic methods and techniques in a broad field ranging from pure mathematics to various applications in physics. The success of the meeting was due first of all to the speakers: thanks to their efforts it was possible to take recent developments into account and to speed up publication of the written versions of lectures given at the Symposium. We are also grateful to the staff of ZiF, in particular to Ms. M. Hoffmann, who expertly handled the organization of the meeting, and to Ms. B. Jahns and M. L. Jegerlehner, who prepared the manuscripts for publication. S. Albeverio, Ph. Blanchard, L. Streit Bielefeld and Bochum, December 1985

CON TEN T S

S. Albeverio, R. H¢egh-Krohn, H. Holden Stochastic Lie group-valued measures and theirs relations to stochastic curve integrals, gauge-fields and Markov cosurfaces . E. A. Carlen Existence and sample path properties of the diffusions in Nelson's stochastic mechanics.

25

A.P. Carverhill, M.J. Chappell, K.D. Elworthy Characteristic exponents for stochastic flows

52

G.F. Dell'Antonio Electric field and effective dielectric constant in random media with non-linear response

81

D. DUrr, S. Goldstein Remarks on the central limit theorem for weakly dependent random variables

104

H. Follmer Time reversal on Wiener space

119

L. Gross

Lattice gauge theory: Heuristic and convergence....................... 130 S. Kusuoka The generalized Malliavin calculus based on Brownian sheet and Bismut's expansion for large deviation 141 J.T. Lewis An elementary approach to Brownian motion on manifolds

158

v J. T. Lewis, A. Truman The stochastic mechanics of the ground-state of the hydrogen atom

168

T. L tnds trem

Nonstandard analysis and perturbations of the Laplacian along Brownian paths

180

Y. Le Jan Hausdorff dimension for the statistical equilibrium of stochastic flows .. 201 H. Naga i Stopping problems of symmetric Markov processes and non-linear variational inequal ities •..••..••..•...••.........•.••......•....•••....••.....•••... 208 M.

Pinsky Mean exit times and hitting probabilities of Brownian motion in geodesic ball s and tubular neighborhooods

216

W.R. Schneider Rigorous scaling laws for Dyson measures

224

R. Seneor Asymptotic freedom: A rigorous approach

234

R. Streater The Fermion stochastic calculus I

245

List of speakers

L. Arnold E. Carlen G.F. Dell' Antoni 0 E. B. Dynkin D. DUrr D. Elworthy H. Follmer L. Gross F. Guerra J. Hawkes Y. Higuchi R. H¢egh-Krohn

S. Kusuoka J. Lewis T. Lindstr¢m Y. Le Jan A. Meyer H. Nagai M. Pinsky W.R. Schneider R. Seneor B. Soui llard R. Streater A. Truman

Stochastic Lie group-valued measures and their relations to stochastic curve integrals, gauge fields and Markov cosurfaces by

Sergio Albeverio

*.11

Raphael Hoegh-Krohn

** II '

,Helge Holden

**, II

ABSTRACT We discuss an extension of stochastic analysis to the case where time is multidimensional and the state space is a (Lie) group. In particular we study stochastic group-valued measures and generalized semigroups and show how they can be obtained by multiplicative stochastic integration from vector-valued stochastic LtvyKhinchin fields. We also discuss their connection to group-valued Markov cosurfaces and> in the case of 2-dimensional "time", group-valued curve integrals. We analyze furthermore, in the general multi-dimensional case, the relation with curve integrals, connections and gauge fields and mention the application of group-valued Markov cosurfaces to the construction of relativistic fields.

*

Mathematisches Institut, Ruhr-Universitat, Bochum and Bielefeld-Bochum Stochastics Research Centre, Volkswagenstiftung

** II

Matematisk Institutt, Universitetet i Oslo, Oslo Universite de Provence and Centre de Physique Theorique, CNRS, Marseille

2

1. Introduction In the case of "one dimensional time", Markov processes on manifolds have been studied in different connections, both by analytical semigroup methods e.g. [1]-[3] (see also e.g. [4]-[6] and for many references [7]) and by probabilistic methods, e v g , [8] - [10] (see also e g , [4] - [7], [11] - [13]). The interplay between i

analytic, probabilistic and geometric problems and methods has been given great attention in recent years (e.g. [14], [15], [7], [16] - [22] and references therein). Markov processes on manifolds have also been used in connection with non relativistic quantum mechanics, see e v g . [6], [7], [15], [29]. The case where the manifold is that of a Lie group is, on one hand, a particular one, but on the other hand, due to the particular geometric structure of Lie groups, it also has pecularities which makes it very worthwhile studying. In fact such studies have been done, particularly in the case of diffusions, see e.g. [10], [13], [22], [23]. An extension of those methods and results to the case where the time parameter lS multidimensional, l.e. of random maps from a manifold (or more generally, any measurable space) into another manifold is of great interest, for many reasons. We mention in particular, besides stochastic analysis itself, e.g. [26] - [28], the study of representations of certain infinite dimens{onal Lie groups and the construction of non commutative distributions, see e.g. [16], [29], the study of non commutative random fields [15] and quantum fields [15], [30] - [32]. In this lecture we shall in particular present a new approach to the construction of the Markov fields in the case where the target manifold is a Lie group. This approach is based on an extension to the case of more dimensional "time parameter" of methods of stochastic analysis on manifolds. It turns out that the random fields which we construct have interesting invariance properties which make it possible to associate to them

relativistic fields and, in the case of a 2-dimensional manifold, random

connections and stochastic Euclidean gauge fields. The basic object underlying the construction of such Markov fields is an extension of the concept of Markov semigroup to the case of index set a measurable space "generalized Markov semigroup" (PA' A E ,13

obtaining so called

) satisfying essentially

3 and a continuity property). Thus in section 2 of this paper we study such generalized Markov semigroups. We show essentially that they are in one-to-one correspondence with stochastic group-valued multiplicative measures 'l on (M,cG) (with 'l(A UB) = 'leA) ''l(B) and 'leA) independent of 'l(B) , whenever An B

=

0, and a continuity condition). As a tool for the cons t ruc t i.oi

of such multiplicative measures we study in section 3 stochastic vector-valued (additive) measures and show that they are classified essentially by a type formula. We use these stochastic vector-valued measures in Sect. 4 to show that, in the case of Lie groups, they are in 1-1 correspondence with stochastic groupvalued multiplicative measures, obtained by solving essentially a stochastic differential equation on the Lie group. Section 5 is concerned with some deterministic notions like multiplicative curve integrals, with their relations to connection and (classical) gauge fields, as well as group-valued (codimension 1) cosurfaces. The corresponding stochastic objects are studied in Section 6. It is shown that they can be realized starting from a generalized Markov semigroup 'l, constructed as in Sect. 3. The Markov property of such stochastic G-valued cosurfaces is also discussed, as well as the relation to stochastic gauge fields in the case the basic manifold M is

• In the case M = R

d

it is also mentioned that to invariant cosurfaces there correspond Markov semigroups on a certain Hilbert space, as well as relativistic quantum fields, associated with hypersurfaces (instead of points). In Section 7 we discuss how the Markov cosurfaces of Sect. 6 can be obtained as continuum limits from lattice group-valued random fields.

4

2. Stochastic group-valued multiplicative measures and generalized Markov semigroups In this section we shall discuss stochastic multiplicative measures and generalized Markov semigroups with applications, in later sections, to the construction of multiplicative integrals. Let

be a measurable space (in the applications (M,d»

ORd ,

(')

If:)

ORd »,

d.

will mainly be

d

(JR ) be i.ng the Borel subsets of JR ) and let G be a locally compact

group. By a stochastic G-valued multiplicative measure n on mapping A E

iA

->-

we understand a

n(A) (w) E G into G-valued stochastic variables, where w is a point

in some fixed probability space wEQ and, when AnB

=

We require n to satisfy n(0)

•

0, n(A) i.ndependen t of nCB) and n(AUB)

A

=

e for all

n(A). nCB), where

A

means equality in law. Moreover we require continuity in law n(A ->- neAl as n) d, d A + A. As an example of n we might think of (M,uS) = (JR .-:3OR » , G = JR, n the white =

n

noise real generalized Gaussian stochastic process on JRd (s.t. neAl = fn(x)XA(x)dx with n(x) Gaussian with mean zero and covariance E(n(x)n(y»

6(x-y».

Most useful for our purposes are stochastic multiplicative measures n taking values in unimodular locally compact groups and having the property that if n(A)

=

e does

not hold P-a.s., then the distribution of neAl is equivalent with Haar's measure on G. We say in this case that n is strongly ergodic. For strongly ergodic non trivial n (L;e , such that for any A E

13

either n(A)

'* e

or n(M-A)

* e)

we have that the

distribution PA of n(A) is invariant in the sense that its density P to Haar measure satisfiesPA(h-\h) see

=

A

with respect

PA(k) , for Haar-a.e. h,kEG. For this result

[39].

We shall now introduce the concept of generalized Markov semigroup on a locally compact group G. We call so by definition any map p from a measurable space (M,A) into probability measures on G satisfying PAU B = PA* PB PA

->-

n

PA weakly as An + A.

By the above, any multiplicative G-valued measure n on

defines a generalized

is the law of neAl. If n is strongly ergodic and non

Markov semigroup on G, p s.t.

t.r i v i.a l then it defines an invariant generalized Markov semigroup PA on G, in the sense that

PA(h -\h)

p/k) for Pr-a s a h , kEG, i

P being

the density of p ,

5

Examples

1. For an example of a generalized Markov semigroup on G, let us consider

the case (M, A)

=

(JR+,

d\ (JR +)).

Let (P , t;;; 0) be a 1-parameter invariant weakly t

continuous Markov convolution semigroup on G, set p[

t ,t 1

2

]

=P t

2-t 1

, 0

t

1

t

2

and extend naturally by continuity the definition of PA to all A E Jj (:nt). Then PA is an invariant generalized Markov semigroup on G. 2. We can use the invariant Markov semigroup P generalized Markov semigroups (M,cB ,a),

provided P +P t

00

of Ex. 1 to construct invariant

t

on an arbitrary a-finite positive measure space

as t +00 weakly (no such restriction is needed if a is

finite). In fact define PA: P a(A) for any A E J3, it is then easily verified that (pA,A E iA) has the properties of an invariant G-valued generalized Markov semigroup. In particular M can be a Riemannian manifold and a the Riemannian volume measure on it, we shall see below (in Sect. 6) the importance of such a case. There is a converse of the above relation: multiplicative G-valued measure

=?

generalized Markov semigroup, in fact we have the following. Let G be a locally compac t separab Le unimodular group. If A + PA' A E on G with parameter space cative measure

on (M,c$) s.t. PA is the distribution of n(A). n is unique in the i.

i)

is a generalized Markov semigroup

then there exists a stochastic G-valued multipli-

sense that for any finite collection A. E tI:3, i the n(A

c!3

=

1, ... ,n, the joint distribution of

is unique. If p is invariant then n is invariant (in the sense that for

any finitepg -measurable partition the joint distribution of the random variables 1.

is the same as that of the random variables

1.

n(A.)g., for any arbitrary 1.

1.

gi E: G). For the proof of these resul ts, based on an extension of Kolmogorov' s construction, see [39]. In particular we get that any strongly ergodic non trivial multiplicative measure is invariant. In the situation of the above example 2 we have easily that, due to the invariance of Pt,PA is invariant und hence the multiplicative measure n constructed from it (unique, up to equivalence, as proven in [39], Theor. 2.15, and such that the distribution of n(A) is Pa(A)) is also invariant.

6 We shall see in Sect. 6 how we can associate multiplicative G-valued integrals to multiplicative G-valued measures, and then use the results of this section to construct the connection and gauge fields mentioned in Sect. 1. However, for the discussion of this association we need first to examine so called stochastic vector valued measures, and this we do in the next section.

3. Stochastic vector valued measures Let V be a real finite dimensional vector space and let (M,dn be a measurable space. A stochastic V-valued measure s

on

is by definition a stochastic V-valued

multiplicative measure on (M,U;) in the sense of Sect. 2, with G equal to the abelian group V. s has then the properties that if An B independent and s (A UB)

s (A) + s(B) and sCAn)

=

example we might take (M,d5)

(JRd,

&'1 (JRd) ),

V

+

=

o then

seA) and s(B) are

seA) in law if An "A. As an

JRs, S the JRs-valued white noise

generalized process. Another example is given by (M,Ji)

OR +, Ii (JR +)

), V = g, the

Lie algebra of some Lie group G, s the white noise generalized process in g. Let us remark that our stochastic vector valued measures are related, but not identical, with those encountered in the literature on processes with independent increments, see e.g. [40] - [46J. It is easily shown [39] that for any stochastic V-valued measure s we have

f

e

for any pEV', A E

, with j1(p;dm) for any pEV' a measure on (M,uS). Moreover s

has a decomposition s SC (A)

= seA n (M-D) ,

\l(p;dm)

A

Sc + sd such that sd (A)

f

l

eA

l

f II all s

(e

s( {x.}) l

are independent. One has

f (p ;m)p (drn)

with f(p,m) of L£vy-Khinchin's type,

0


and

the configuration process is a

and bacKwaed

in caE-e

is t'lar-Kovi an .

-11*

ar-e (possibly time dependent)

elliptic oper-ator-s of

the form

For any f E C:(/Rr»

and T 6 /R:

to

is a

martingale on

(12.)

T

and

T is a

3-t

I;

i-

martingale on (-oo,TJ.

Note that the definition doesn't include any explicit regularity

31

condi t ions on the oper-a tor-s

and

. This

is intentional, but when

we ar-e dealing with coefficients as singular- as those discussed in the fir-st section, we must explain what we mean by elliptic in (ii). Let

p(Jx,'i)

be the image of Pr- under- E;(t);

say that

ts

this is a measur'e on

fRtI.

When we

is elliptic, we mean that almost ever-ywher'e with r'espect to

this measur-e,

lb(x,t>!
0 with its standard embedding

TxSn(r),

E

n-l -2- < u, v>, a x (u , v)

Ric (u,v)

r

r

x

'r

whence n

1

- 2" 2" r

Thus the gradient Brownian exponent

on Sn(r) is

to the

of one

operator of Sn(r).

because the mean exponent

with the maximum the

In fact there are no other exponents 1

n Al: is also the leading eigenvalue: see

§3 below.

3.

MEAN EXPONENTS

A.

Baxendale pointed out to us that there is a formula for the

weighted sum of the exponents A[ ,

(see §O.C), which needs only know-

ledge of the invariant measure p of A and not those of any derivative systems.

Using this together with results of Reilly [18], Chappell [8] m was able to show that for any compact M embedded in R the induced gradient Brownian fLow satisfies: the mean exponent

B.

Al:

s the l.ea d i.nq e i q enva Lu e of

/::,

(l3)

From the proof of the multiplicative ergodic theorem, [19J and its

extensions [2], [3] to our situation we have

for p

F-almost all (x,w), where the determinant is taken using the

Riemannian inner products on the tangent spaces at x and

(w,x) •

Thus Al: has the geometric interpretation as the exponential rate at which the flow changes volume (or area if dim M = 2).

63 Ito formulae for the determinant have been obtained by various authors, in particular Malliavin [16J. that the S.D.E.

An easy way is to observe

(2) for the derivative flow determines an S.D.E. on

the manifold L(TM;TM)

L(T M;T M)

x

Y

consisting of the disjoint union of the spaces of linear maps of TxM into TyM.

The precise form of this S.D.E. is irrelevant.

We then

have the smooth function Idetl:

lL:is

(TM;TM) .... R

on the open subset of L(TM;TM) consisting of invertible elements.

If

we apply the form of the Ito formula given in Lemma VII 9B(ii) of [9J, use of the classical continuity equation for the Jacobian determinants of flows of ordinary dynamical systems quickly yields

(14)

which gives Baxendale's formula

= When A

=

fM div A(x)p(dx)

+

f

M

r: (dx) (15)

i.e. for a flow of Brownian motions, p(dx)

=

IMI-1dx

where IMlis the volume of M and we can integrate by parts and apply the divergence theorem to obtain 1

- 2TMT

i

(div Xi(x»2 d x

(15 )

It is not clear to us whether, for an arbitrary compact Riemannian manifold M, it is possible to choose x

1,

••• ,x

m

with div Xi

=

0 for each

64

for a suitable choice of A.

i, and with A =

is simply that for each x

This last condition

M the corresponding X(X):R

m

+

be surjective and induce the given inner product on TxM.

T M should

x

If not one

can ask what the maximum value of AI: can be over all such choices of 1

X , ••• ,X

m

I

for m allowed to vary, and whether this number has signi-

ficance in other directions. A corollary of the Hodge decomposition theorem extends Helmholtz's theorem from the case M = R

3

to enable us to write xi uniquely as the

sum of a divergence free field yi and a gradient, [13J; xi say, for

o} the associated semi-group.

The following are applications of the regularity theory and the strong maximum principle for solutions to problems involving the operator i)2 A= (X + Ai the first is standard, and the second comes from 1

[4J (see Remark 2.43 there) where there is a detailed proof. (i)

There is a unique probability measure

=

t

>

O.

=

Moreover

on M such that

Ao(x)dx where Ao is a positive

element of C""(M) (ii) Given {o*, •.. ,om}

c

C""(M) , set

y

If Y

=

0, then there is a unique f

Xi f = 0 i' 1

B.

; ...

E

C""(M) such that

ff

o

and

m.

For given elements

0

1 , ••• ,0 m and Q of C"" (M), define

t Oi(x )dB i + Jt Q(x )ds s s 0 s

Jo and define y as in (ii).

I(p)

sup inf

¢

I f Y > 0, define I:R ... [0,"")

by

(p-J (Q-A¢)d\-l)2

J

21: (oi_Xi¢) 2 d\-l

taken over \-l in M1 (M), the probability measures on M, and ¢ E C""(M). 2/0 Here and below 6 is interpreted as + "" if S 0 and as 0 when [3 = O. If Y

0, let f be as in (ii), set Q

I:R ... [0,"")

u

{""}

by I(p)

inf

Q - Af, and define

E

M (M)

1

&

p

JQ

74 where Coo(M)}.

sup

The following summarizes some of the principle results of [4J set into this context, as described in Remark 2.44 of [4J. Theorem convex, c(p-

(Q

(2y)-1

(Stroock [4J):

I(J

=

Q

The function I is lower semi-continuous and

0 and there is an c > 0 such that I(p)

(p -

JQ

Moreover: if y > 0, then I(p)

R.

for all p

for all

p

R, and if y

tinuous on (Range Q)o and takes the value +

00

=

0 then I is con-

off of Range Q.

Fin-

ally for any Borel set r of R:

-

inf I

lim

O

T+oo

r

lim T+oo

1 log

inf X M o

T

sup X M o

T

1

]E'

(p(T,X o) IT

log :P (p (T ,X o ) IT

r: r:

s - inf 1.

r

In particular, if

r

r

O

and either y > 0 or ar n a(Range Q)

then

o.

1/

Now furnish M with a Riemannian metric and associated LeviCivita connection and set

Then, from equation (14): Corollary:

Set oi

=

div xi, 1

ism, and

¢,

75 m L

;z1

div A +

i i .

1

If I is defined accordingly, then the preceding applies with P (t ,x'l

II When

Remark 1 (i) becomes:

m

the expression for I simplifies and

- JUdJl) 2

inf u

I (p)

o

L cixi i=1

2 LOi dJl

The proof of this comes from equation (2.14) of [4J, the expression given there for A(A)

in (2.25)

the convexity of Jl

I

+

J(jl)

I

and the mini-max theorem. m 2 _ (ii) If 11=

1

4"

nu

M.

For other immersions A < nj.! and so if f is an immersion, not

E

of the special class, but with mean curvature of constant length, then lim

t..-""

[lJ

t

10gP {!det T

Arnold, L.

Xo

(1983).

Ftl> 1}

A formula connecting sample and moment

stability of linear stochastic systems.

Report 92, Forschungs-

schwerpunkt Dynamische Systeme, Universitgt Bremen.

Also article

in these proceedings. [2J

Donsker, M.D. and Varadhan, S.R.S.

(1975).

Asymptotic evalua-

tion of certain Markov process expectations for large time, I. Comm. Pure Appl. Math., 28, 1-47. [3J

Stroock, D. deviations.

(1984).

An introduction to the theory of large

Universitext Series. Springer-Verlag.

80

[4J

Stroock, D.

(1984).

On the rate at which a homogeneous diff-

usion approaches a limit, an exercise in the theory of large deviations.

To appear in Ann. of Prob.

ELECTRIC FISLD AND EFFECTIVE DIELECTRIC CONSTANT IN RANDOM WITH NON-LINEAR RESPONSE G.F. Dell'Antonio*) Dipartimento di Matematica, Istituto G. Castelnuovo Universita di Roma - La Sapienza and International School for Advanced Studies, Trieste, Italy Introduction In this talk I will outline some results on the existence and the average properties of a solution of the time-independent Maxwell equation in a non-homogeneous or rancom dielectric

medium with non-linear

response. I shall also discuss some approximation schemes, and prove in the random case a point-wise ergodic result. Details of the proofs ann further results can be found in [1], [2]. In particular in [1] we prove that the formalism I use here can be adapted without essential changements to the study of other physical problems, e.g. thermal conductivity, magnetic susceptibility, elastic response, velocity of sound, viscous flow. In fact, all these problems are mathematically equivalent from the point of view considered here. The content of this contribution is the following. In Section 1 I describe the formalism in the case of an inhomogeneous medium and outline the proof that at least one solution exists when the non-linear terms are sufficiently small. In Section 2 I consider the case of a random medium, and show that it is mathematically equivalent to the one discussed in Section 1, apart from a difference which is pointed out. In Section 3 I discuss some properties of the effective dielectric constant and some approximation schemes. In Section 4, for a particular geometry and under an ergodicity assumption, I prove that the infinite-volume limit, for the electric fieln and for the effective dielectric constant, exist and are configuration independent. In conclusion, I am glad to express here my thanks to Profs. S. Alheverio, Ph. Blanchard and L. Streit for the invitation to this Conference and for having provided a stimulating scientific atmosphere.

* ) CNR, GNFr1

82 1. Inhomogeneous Dielectric Media with Non-Linear Response We denote by

D

R3

a bounded domain in

of dielectric tensors. We assume that

D

sumption can be easily lifted [1]. We write

e (E) The parameter

A

(x)

= s

«» (x )

and by

s{E) (x)

a field

be simply connected; this as-

+ A(j){E) (x},

s{E) A

0

as (l.1 )

•

measures the strength of the non-linearity. We shall

soon specify our assumptions on the dependence of

(j)

on

E.

We want

to find solutions of the equations rot E

o

(l. 2) a }

xED

o

div £{E)·E

{1.

2)b

under suitable boundary conditions. This will be achieved by generalizing the method of orthogonal projections, introduced in [4] and further developed in [2] for the linear case. Since

D

is simply connected, equation {1.2)a implies that one

can find a potential field

cp

such that

E = Vcp.

Equation {l.2)b

takes then the form div s (VCP)

• Vcp

o .

(l. 3)

(V,CPo) where V is a closed proper subspace of (D) such that V.:2 and 0

we write

Q

AQ E

E 0

1;*

in

. When 0

(3.2)

where A = 0

c

and

Eo

is the unique solution of (1.2) a, b

for

96 From the first resolvent identity one has (I - Q

E

( A) AQ

o

( A))

E

(I _ Q AQ )

1

c

0

From (1.18) one can now determine

E

c

1

+ 0 ( A) •

(3.3)

in the following expansion

1

E(A)

If the functional ated

k

(3.4)

W(E)

is of class

c

K

times; in particular, if

E)

depends polynomially on

this procedure can be iteris of class

COO

(e.g. if it

the iteration can be continued indefinite-

ly. One proves then [2]. If

Lemma 3.1

is of class

COO

the solution

E

of Maxwell's

equation, described in Proposition 1.6, has an asymptotic expansion in powers of for all

i.e. there are functionals

A,

M

R

,

i

L = 1,2, •••

r

such that

1 (3.5)

E (A)

The first new functionals

are constructed in [2]. From Lemma 3.1 i one derives easily the corollary: Corollary 3.2

Under the assumption of Lemma 3.1, the effective di-

electric constant are constants

R

c

has an asymptotic expansion in

£*

n

n that, for every integer £* (A)

which depend only on

1

N

B,A

A, i.e. there and

such

£ (0)

1 N

r

+

£* (0)

(3.6)

k=l

An important special case is obtained if

E (0)

takes only a finite

number of values

E1 ... E Consider for definiteness the case M. One can prove [1], [2] that £*(0) and all the cn's

Em = x

I. m• (3.6) are boundary values

of functions

*

t: (0) (z I •.. zM) ,

n

which are analytic in a domain smallest connected domain in a)

e

hyperplanes"

de

e

is the

.i = 1. .. M}

is contained in the union of the "half i j = 1 ... M defined by

r .. , lJ

defined as follows.

with the following properties

{zlzi = i ,

contains the point

b) The boundary

e eM

L, •.•

'"

in

97

This proof makes use of the structure of Equations (1.18) and (1.22). For

£*(0)

the result follows also directly from the elliptic charac-

ter of (1.2) a, b Using Weyl's theory of analytic polyhedra, one can give a representation for functions analytic in

8. This representation together with

(1.18) could be used to provide estimates on known for some special choices of

£*

when its value is

. x ... x M 1

4. The Infinite Volume Limit

In this section we shall study the relation between the results of Section 1 and those of Section 2. We will do this for a special choice 3; of domain in R more general cases can be treated similarly. We shall follow closely the method of proof in ([5], Appendix); the results we obtain are stronger, even in the linear case. The problem is the following: as in Section 2, let

be the col-

lection of all possible realizations of the dielectric medium. All realizations are assumed to be periodic of period one along two orthogonal 3

directions in R , so that they can be viewed as tensor­valued fields 2xR on T We denote by (x a generic point of T 2xR. 1,x2,z)

w

Each realization E .... £(0)

from vector fields on

provides then a map

+ A­

A

Jr S(x 1 , X 2 , L

00

Z 7W) d s

x

we define

o

(4.1 )

in

­L

is ergodic if the following assumption is

We shall say that satisfied for all

S

of the class described above,

We shall also need an assumption on the dependence of

0,

be the solution of

EL(X,w)

c(E) (x,w)

with dielectric tensor

VU,

Denote by

let

U(L)

and bound-

°.

U(-L)

(4.2)

the s.s. random field solution of (2.7) with bound-

E(x,w)

ary condition

I

bk

(4.3)

With this notation, the assumption we make on a

2:

There exists a function

y: R+

x

R+

c(E)

is the following infinitesimal at

+

the origin in the second variable, such that

where

.i.

Ilf (.

2

T x[-L,L]

2L

.

(4.5)

One can then prove Proposition 4.1 Q

o

c

Q

Assume

such that

.i

lim

Proof

a

and if

2

There exists then a subset w E Q

o

o .

(4.6)

Let Q

1

1

By assumption For

and

l = 1

2 IE(x,w) - EL(x,w) I dx

2L

L+oo

a

w E rl

1

{wi lim 2L Le-cc

al'

I

L

°

-L

=

in

(4.7)

1

define

z

J

-z

I

1

°

'E(r ,O;w)dr

(4.8)

99

where

and

It is straightforward to verify that

I7U Moreover, when

(4.9)

E - bk

E

W

1. L U(x,L;w)

+

0

Let

J U 2 (2i, L ; w) d X

B(L,W)

(4.10)

T2 and remark that

B(L,w) We define

L +

0

if

W

E

(4.11)

.

00

as follows L

1L {to l Ldm 2

L+oo

J[E 2 (2i, S ; w) Jr

E

2

(w) u (dw) Ids

-L

=

Let Let

n

w E

(4.12)

by assumption

=

lJ

1

•

we shall prove that one can find [5] a sequence of domains

VN(w) :

and a sequence of functions

XN(z,w)

of class

c1

in the first vari-

able, with the following property

o

a)

b)

1

!VNI

J V

(1

if

Iz I

2

XN)E (x,w)dx

N+oo

LN (w)

0

N

c)

1

TV;!

J V

N

(I7X

N)

2

2

U (x,w)dx

N+oo

• 0

(4.13)

100

Assuming for the moment that such sequence of functions has been constructed, we complete the proof of Proposition 4.1. We shall treat separately the linear and non­linear case. 1) Linear case. Let One has

UL

be a potential for

2L

--L --L s

2L

f v

with

UL(O,w)

[ (IIU ­ IIU sO (IIU L) L

UV (1 ­ XL)

IIXLU

s+ --L s 2L

f V

1dx

::>

(4.14)

L

::>

0.

f

--L

::>

EL(X,w) ­ bk,

I VUL ­ VU I I( 1 ­ XL)VU +UVXLldx

L

In deriving (4.14) we have used (1.2)

a,

b'

(2.7) and the inequality

S'I::>S(O)(X,W)::>S+'I.

From Schwartz's inequality one has then (l

so that, using a), b), c), if

w E

2) Non­linear case. Proceeding as in case 1), one obtains

JI V L

--L 2L

From (1.1) one has

J I VU·

(I: (E)

VUL ­ VU I I VU ( 1

(4.16)

101

so that

It- 21L J IVu· (E:(E)-E:(EL»V(U-UL) IdxSAIIElll1

II\.P(E)-\.P(E L) 11=.

IlL

V L

(4.17)

a

From (4.16) and (4.17) together with assumption

it follows

2

(1 - II E II y(1I E - E II L' A)jIE L (4.18)

[2i J (VU) 2(1 - XL) 2d x + 2i V L From assumption

a

and the definition of the domain 1 that the right hand side of (4.18) converges to zero when Moreover, one can find a positive constant stant

A o

it follows L

+

00

•

(smaller than the con-

in Proposition 1.7) such that

(notice that

E

an upper bound on

and

E

liE Ii

L

depend on and

We conclude that, i f

IIE L ilL

but Proposition 1.7

A ,

for

A E [0, A 2]

and

provides

A E [0, AI) w E

0

It remains therefore to prove that one can construct ([5], Appendix) a sequence of functions

X N

with the properties

a,b,c

described in

(4.13) •

For each

w E

XL

by

s

XL (s,w)

1 -

s L(1-6(L,w»

L(l

XL(s,w)

1

-L(1-6(L,w»

s s s L(l

of class

bourhood of

XL(s,w)

°

XL(s,w)

Let

we define

o '

±L

=

co

C

and of

= ±L 6(L,w»

be obtained by rounding off ±L(l-S(L,w»

.

S lsi SL 6 (L,

X (s )

(4.19)

to ) ) •

in a neigh-

102

(4.20) It is straightforward to verify that

XL

satisfies

a), b), c)

in

(4.13). This concludes the proof of Proposition 4.1. From Proposition 4.1 one derives an ergodic result on the effective dielectric constant. We formulate it as Proposition 4.2

Let

EL(X,w)

t*

and let

tt(w)

be the effective dielectric constant for

be the effective dielectric constant of the ran-

dom case. One can find a set

Q

3

c

Q

t*

1

Proof Q

Define

Q

4

1 2L

4

with the properties

by L

f [(T Sk E(E)E)

(to ) ­

­L

I

(E(E)E) (to l u f dco ) ]dS

oI (4.21)

Let

and notice that

in view of assumption

a

.

l

Define

:t(w) -

2L

f

(E(x,w)E(E) (x,w)k)dx

VL Since

Q c 3

Q4'

one has

lim (Et (w)

L+

-

E*)

00

Let sup I E (E) (w) I , w

(4.22)

103

Notice that Since

11

c

3

c 11

2


- 00

s

-

lim L->- 00

+ lim L->-

so that,

II E - Ell L ... 0

00

---L 2L

I

I

I ((E (x,w)

-EL(X,wȣ(E) (x,w)'k) [dx

I

I (E L (x , w)

(£ (E) - £ (E

(4.24)

V L

---L 2L

V L

using Theorem 4.1 and Assumption lim I -: (w) L->- 00

(w)

I

o

vw

L)

) (x , w) -k ) Idx

A'

2

E 11

3

(4.25)

.

Proposition 4.2 follows now from (4.23) and (4.25).

[]

References [1] G.F. DELL 'ANTONIO,

R. FIGARI, E. ORLANDI, Representation and asymptotic expansion for the electric field in random or inhomogeneous media. University of Marseille preprint, June 1984.

[2] G.F. DELL 'ANTONIO, Existence and representation theorems for nonlinear electrostatics in inhomogeneous media, of Bielefeld preprint, October 1984.

ZiF, University

[3] F. GILBARG, N. TRUDINGER, Elliptic PDE of second order, Springer Verlag Berlin-Heidelberg-New York 1977.

[4] H. WEYL, Duke Math. Journal 1-, (1940). [5] K. GOLDEN, G.

PAPANICOLAU,

Comm. Math. Phys. 90,

473

(1983).

REMARKS ON THE CENTRAL LIMIT THEOREM FOR WEAKLY DEPENDENT RANDOM VARIABLES Detlef Dilrr*, Sheldon Goldstein§ Institut des Hautes Scientifiques 91440 Bures-sur-Yvette (France)

I. Introduction In recent years the Central Limit Theorem (CLT) for martingale differences [1,2]has been frequently applied to stationary sequences {X.}. l

l

b

to establish the convergence of the normalized sums

1

/i1

1

n-1

/Ii

i=O

L:

X.l

to a normal law. A sequence {mi}i Z of random variables adapted to some increasing family of a-algebras

are called martingale

differences if E(mi+1ITi) 0 for all i . The first one to observe that the standard results for CLT' s for weakly dependent random variables may be obtained from the CLT for martingale differences was Gordin

DJ.

This approach was then taken up

by others (see the monograph by Hall and Heyde and references therein

[4J i. The idea is to approximate Xi by a martingale difference mi in

such a way that the "error"

1

/i1

in probability. Then one obtains easily that S

1

same law to which

n -1

1:

converges.

n

/111

converges to the

Itl i=O In these notes we derive, using a naive and simple approach, very weak conditions under which a stationary ergodic sequence may be approximated by a stationary ergodic sequence of martingale differences

Supported by a DFG-Heisenberg Stipend, present address: BIBOS, Bielefeld, 4800 Bielefeld, West-Germany § Research partially supported by

NSF Grant NO PHY-8201708; on leave

from Department of Mathematics, Rutgers University, New

08903.

Brunswick,

105

From this we obtain immediately the classical CLT's for weakly dependent random variables. For completeness we review in the next section the classical results. Section III deals with the approximation and contains our main result, a direct consequence of a simple (abstract) Hilbert space lemma,proven at the end of this paper in section V. In sectl.on IV we generalize the approximation procedure to situations which frequently appear in applications, namely when Xi rable w.r.t. the a­algebra

7.l ,

is not measu-

which is the one used to define the

martingale difference mi' II. Classical results Let {Xi}iEZ be a stationary sequence of centered square integrable random variables defined on a probability space The classical conditions under which a CLT for the normalized sum 1

(2.1 )

IiJ

S

1 n

IiJ

n -1

Z i=O

X.l

holds, are in the form of mixing conditions (weak dependence of past and future) which enables one to reduce the problem to a CLT for in­ dependent random variables: Let

=

a(Xj,j ( 0)

denote the a­algebra of the past up to time 0 and

:t n

=

o(X.,i :> n ) .1 .1

that of the future after time n. Two standard mixing properties are

I:

n=O

IIE(xn

0

)11


4 there isn't a converges to something good heuristic basis for believing that 1J that deserves to be represented by (2.E). But the case of real

139

physical interest is d=4. It is widely believed that in this case and with G = SU(N) one should let the coupling constant c in (3.6) go to zero along with s at just the proper rate that the ) will then converge to a (perhaps generalized) measure which represents the true meaning of (2.6), with c = O. The circle of ideas involved with letting c(s) + 0 goes under the name assymptotic freedom. The following theorem shows that in case d=3 and G = U(l) the preceding heuristic discussion is justified. Write

Then in view of (3.3) approximation to

(x)

J

should be reoarded as the lattice

Fjk(x).

Theorem 1. Let d=3 3 2-form on R of class F (p)

c:.

and

I

G Put

j 0,

t

We assume

Lie[V 1 , · · · ,V d] (x O) Then the probability distribution of X E has a smooth density 1

ply} = p(s,xo,y), s = E

2

lE:: dy ] = P[

Since P[

X

1

E (; dy]

= p(s,xO,y}dy, p(s,xo,y) is the transition probability density function of a diffision process whose generator is L

=

1

d

Z V.

2

2 ill

Our concern is in the asymptotic behavior of P(s,xo,y), s + O. other words, we want to know the asymptotic behavior of P[

In

X EE dy 1

as E + O. Let 8

{ e : [O,lJ

+

N; R continuous, e(O)

is a measurable function F : 8 think that X1 E(8) F(e)

=

y).

Hypothesis 1.

=

+

R

N

with

= 0 }.

x1 E ( e ) =

F(Ee} with probability one.

F(e).

Let M

Then there We may

={

8

8;

We assume also the following. There exists a unique h

denote by H a Hilbert space ( e f

8;

O

such that


0, are mutually (2)

How one can justify Schilder type theorem for such a

non-continuous functional on 8?

Taylor's expansion does not apply

in this case. (3)

Since we restrict the functional F on TO ' we have to be

involved in noncausal stochastic integral. integral is not convinient tool.

But noncausal stochastic

Can one avoid it?

Bismut[2] considered concrete functionals, solutions of S.D.E., and so he did not have to think of the definition of F(Ee). not avoid to think of noncausal stochastic integral.

He did

He made effort

in giving the definition for z{u) and in showing Schilder type theorem.

We take completely different strategy from his.

think of d-dimensional Brownian sheet WS{t)

W( s,t)

{Wi(S,t)i i=1, ... ,d} and think of functionals of Ws• let us think of S.D.E. d

\

I Vi(X(t» i=1 X(OiS,W

S

)

= x

dW

s. 1

(t)

+

First we

s VO(X(t»dt

For example,

145

Then functionals X(liS,Ws), s probabilty P[ X(liS,W

s)

0, are our object.

Since the

dy ] is equal to the transition

N probability p(s,x,dy) of the diffusion process on R with a 2 th e asymp t 0 t'lC . . -1 d v.l + V 0 ' our concern lS In 2 i=l behavior of functionals as s + O. We will show Bismut's expansion

generator L

formula for general class of functionals in the following way. Step 1.

We construct stochastic ana

is including generalized

Malliavin's calculus for functionals of a Brownian motion on an abstract Wiener space.

Then we will get stochastic Taylor

expansion. step 2.

We show an abstract Schilder type theorem by relying on

Wentzel-Freidlin type argument and stochastic Taylor expansion. step 3.

We show that our class of functionals are robust even if we

split the Hilbert space and the Brownian motion as in the case where H = TO e NO'

Then we will get Bismut's expansion formula

automatically. Finally let us make some remarks on the advantage of our method. First of all, we can think of various kinds of asymptotic problems at the same time.

For example, we may think of the asymptotic

behavior of X(l is,Ws) where X(tis,W

\

dX(tis,W

s)

=

d V. (X (t))

i=l

1.

dW

s)

s. 1.

satisfies

(t)

+

We can handle Ito ( non-Markovian ) process. the case where the set { h Eo Mill h llH

inf { \1

We can also think of

e II H i e

t

M }} makes a

finite dimensional manifold, but we will not discuss this case in the present paper. 2.

Generalized Malliavin Calculus.

Let B be a separable real

146

Banach ( or

) space and H be a separable Hilbert space for

which H is continuous

and densely embedded in B.

dual space H* of H with H itself.

We identify the

Then the dual space B* of B can

be regarded as a dense subspace of H.

Let Us' s > 0, be Gaussian

measures on B such that

r

} B

exp( 1-1 B

B* •

for any u

We will write W or W for the totality of continuous functions from B [0,00) into B.

Let P

P

be a probability measure given

by P[

w(t 1 ) E A1 , wlt i

n II

2)-w(t 1 1

u

A ... , wlt ) 2, n)-wlt n_ 1

)

-1

An

(A. ) l

We will call the triple (H,B,P) an abstract Wiener process.

This

object was first studied by Gross[4]. For any separable real Hilbert space E, we set FC

OO

{

f

t

(

[

t

0,00) x B; E) Coo([O,oo)xB;E); there are n

functional A : B 'f(s,Az),

+

R

n

1, a continuous linear

and

(s,z) Eo [O,oolxB ).

such that f(s,z)

=

Here

;E); for any multi-index a, there are C O.

•

- 21 II h IIH 2 i

N L

exp(

L


-

f(s,hO+wO(s)+k).

Let Ur

loc(K->-E,dk)

' r > 0, denote

Then we have the following. > 0 and integer n, if f

Moreover, i f

iE),

then

is completely regular,

154

then

is also completely regular. Now we can state Bismut's Expansion Theorem. N. Let y to R

Theorem 5.2.

completely regular.

(A

Suppose that FE;

;R

N)

Suppose moreover

(1) lim slog E[ det{ (DFi(S,W(S),DFj(S,W(S))lD(R)}'

s+ 0

(2 ) h h

'"

(3 )

O

< p
0 and E > 0

2 _ < C (U )

E

}

-+

r

= y, w(k) t y if k E Ur\{z(w)}, and

i f k IS U r

1,2, be

U r/ 4

155

smooth functions and 0 < °1

II w-w oII HN+2(ij

°1

if

II

°2

if

II W-w OII

if

o 1 ( w)

=

0

P2(W)

Let ¢

satisfying

< °2

0RN) 2 r' w-w OI N+2 - R) N H2 (Uri

and

N+2 N < 2° 2 (Uri R) H2 i f Ilw-wol\ 2 N > E/2 C (UriR )

0

Coo(KiR) satisfying

¢(k)

= \

10

Now split H to HOeK, and let B

O'

if

[k ]


r

wO(s,w), ... as before.

Then

p(s,y) E E

P P

[ o( F(s,w(s)) - y ) ] [

1 - ¢(P Kw(s)-h F(s,hO+wO(s,w))) O)-P 1( +

1

1

(s)

P

E [ ¢(PKw(s)-h + 1

2

(S)

O)

)-o(F(s,w(s))-y)

-P ( F(s,hO+wO(s,w))) -O(F(s,w(s) )-y) 1

]

•

Then for the second term 1 ( s ) , we have 2

(_1_)N/2 exp( _ 1 2ns 2s P x

E

II h II

2

0 H

r

o[ J dk exp( K

x P ( F(s,hO+wO(s)))-o( F(s,hO+wO(s))(k)-y) 1

( 1 ) N/2 ( 2ns exp where

f(s,wO(s))

2 2: \I h ollH ) E

PO[

g(s,wO(s)) exp(

f(s,wO(s)))],

156

¢>(k(s,wO(s)))

-° 1 (

F(s,hO+wO(s)))

x detK(D K F(s,hO+wO(s))(k(s,wO(s)))

* -1/2 ,

DK F(s,hO+wO(s))(k(s,wO(s)))) and k(s,wO(s)) = P2( F(s,hO+wO(s)))-z( F(s,hO+wO(s))) . Then from the assumption and Theorem 5.1, we have k t f,g E

g(.l0iR),

Df(O,O)

and they are completely regular.

aO. Let 9 be an arbitrary C2 - f unc t i on g: W .. R, and apply Ito's formula to the process g(X): (3.2) From (3.1) and (1.1) we have P(Xt)dt,

(3.3)

so that d 9 ( Xt )

dNt + j(Xt)·(grad g)(Xt)dt +

trace [g"(Xt)P(Xt)]dt

(3.4)

where ( 3.5 )

Thus

(3.5)

162

where Nt is a continuous local martingale; we conclude that is the generator of the diffusion X. It remains to show that X remains on V=rl(c)fort>O. Letg fj,j=I, ... ,r; then

since (grad d)(y) is orthogonal to Ty, and

for the same reason. It f o l l o ws from (3.4) that dfj(X t) Thus X stays on V = f- l (c) for t>O since it starts there.

0 for j=l, .. ,r.

Remark: The equation (3.1) for Brownian motion on a submanifold of Euclidean space was given by Baxendale [6]. §4.

Martingale Characterization

The description of Brownian motion on V the f o I l ow t nq

f-l(c) given in §3 suggests

Martingale Characterization of BM(V): A process X on ffid with f(XO) = c is a BM(V) if and only if X is a semimartingale such that (1) (2)

dXt - j(Xt)dt = dt1t, where dt = P(Xt)dt.

is a continuous local

We have to show that, given a semimartingale on ffid satisfying (1) and (2), the r e ex i s t s B, a 8M(ffi d l , sue h t hat (4.1)

Let

B be

a BM(ffid) which is independent of X, so that 0,

(4.2)

and let g be a process on ffid such that 80=0 and (4.3) then by (2) and (4.2) we have

163

1 dt .

(4.4)

It follows, by the martingale characterization of BM(ffid), that B is a BM(ffi d) and, by (1), t hat

It remains to show that P(Xt)dMt

dMt.

Consider the process N on ffid

such that NO=O and (4.6)

Then (4.71

so that NN T is also a continuous local martingale; but NN T is nonnegative so that NN T is constant almost surely, and so dNt=O and (4.8)

§5.

Examples

(1)

Hypersurf aces in ffid In thi s case, r=l and j (x)

d­l

-2-H ( x ) n ( x ) ,

(5. 1)

where H(x) is the mean curvature of V at x, and n is the orienting normal vector field. Then BM(V) in the ambient Euclidean space i f and only i f the mean curvature__of V vanishes (Compare [7]). It follows from (3.1) that, if X is such that ( 5 .2)

then X is a BM(V). It follows from the martingale characterization that an alternative equation for BM(V) is

where B is a BM(so(d)), a Brownian motion in the Lie algebra of the

164

orthogonal group SOld), since dt P(Xt)dt; see [21. (2) The unit sphere S2 in we take nIx) x, In the special case of S2, the unit sphere in the outward normal at x; then the principal curvatures are both equal to -1, so that j(x)=-x. The projection PIx) onto the tangent at xis g i ve n by P( x ) = (1 - xxT l . The n (5. 2) y i e 1 d s the e qua t ion 0 f St roo c k [7 ) (5.4)

On the other hand, (5.3) yields the equation of Price and Williams [1]: dXt

+

Xt dt = Xt x dBt·

(5. 5 )

Curves in Let s + x I s ) be a C2-curve in parametrized by arc length; the tangent vector tIs) at x(s) is given by (3)

dx(s) ds

t (s )

then

(5.6)

and i!.(s)

ds

= kIs l n t s )

(5.7)

where n I s ) is the principal normal at x I s ) and k t s ) is the curvature. Then j (x ( s ) )

k ( s )n ( s ) ,

(5.8)

and P(x(s)) = t(s)t(s)T. Now let b be a beginning at x(D) and

and put Xt

(5.9)

x(b t).

Then X is a process in

(5. 1 D)

so that (5.11)

It follows from (5.11) that

165

( 5. 1 2)

using (5.8) we have dt

P(Xt)dt.

By the martingale characterization, it follows that X is a Brownian motion on the curve s + x Ls ) • §6.

Martingale Representation

Let X be a Brownian motion on V be defined by YO=O and

f-l(c) starting at x, and let Y

(6.1)

dYt so that dY t is the tangential component of dX t. Let Brownian motion on V = f-l(c) starting at x, and let

X be another

Y be

defined by

YO=O and (6.2J

dYt

Suppose that X is adapted to the filtration of X; then we have the following Y and

Martingale Representation: Ita equation

Y are related by the (6.3)

where (1) for each

t ,

Ct is an orth

ransformation such that (6.4)

for each unit normal vector field n on V. (2)

the process C is X-predictable.

Let I n j , V; let (b l, of both X and

,n r} be an orthonormal set of normal vector fields on r} ,b be a set of independent BM(R l ) - processes independen X so that (6.5)

and

166

Then, by the argument in §4, the processes Band B such that BO=BO=O and (6.71

are both BM(ffid) and X and X satisfy dXt - j(Xt)dt = P(Xt)dBt, dXt - j(Xt)dt

(6.8)

Moreover, B is B-predictable so that, by the martingale representation theorem for BM(ffid), there exists a B-predictable process C of orthogonal transformations on ffid such that (6.9)

Hence, from (6.7), we have CtdYt +

r

.

< Ctnj(Xt)db J

j =1

(6.10)

Forming the bracket process of both sides with the process b k, using (6.5) and (6.6), we have

establishing (6.4), and (6.3) follows by subtraction. (6.4) that C can be c ho s e n to be x-oredtct.eb le ,

It follows from

Special Cases: (1)

For a hypersurface (r=l); taking n to be the orienting vector field, the map x -.. n I x ) is the Gauss map. (2) Specializing to 52, the unit sphere in ffi3, we have n(x)=x and we recover the result of Price and Williams [1]: Let X and X BM(S2) - processes x ; suppose that X is adapted to the filtration of X. Then the increments dY and dY are related by the Ito equation dYt = CtdYt where

(6. 12)

(1) for each t, Ct is an orthogonal transformation such that (6.13) (2)

the process C is X-predictable.

167 Acknowl ements It is a pleasure to thank Michiel van den Berg and Paul McGill for many stimulating discussions. References

[7]

G.C. Price, D. Williams: Rolling with 'Slipping': I. Sem. Prob. Paris XVII. Lect. Notes in M ths. 986, Berlin-Heidelberg-New York: pr ger M. van den Berg, J.T. Lewis: Brownian Motion on a Hypersurface, Bull. London Soc. (i n press). J.T. Lewis: Brownian Motion on a Submanifold of Euclidean Space, (preprint: DIAS-STP-84-48). K.D. Ellworthy: Stochastic Differential Equations on Manifolds, LMS Lecture Notes 70, Cambridge: C.U.P. 1981. N. Ikeda, S. Watanabe: Stochastic Differential Equations and Diffusion Processes. Amsterdam-Oxford-New York: North Holland 1981. P.H. Baxendale: Wiener Processes on Manifolds of Maps, Proc. Royal Soc. Edinburgh 87A (1980) 127 152. R.W.R. Darling: A Martingale on the Imbedded Torus, Bull. London Math. Soc. 15, 221-225 (1983).

THE STOCHASTIC MECHANICS OF THE GROUND-STATE OF THE HYDROGEN ATOM A. Truman University College of Swansea Singleton Park, Swansea SA2 8PP Hales J.T. Lewis Institute for Advanced Studies 10 Burlington Road, Dublin 4 Ireland

Dub 1i n

Talk given by A. Truman at the BiBoS - Symposium Bielefeld, September, 1984. §

1.

I ntroduct ion

The prospect of a formulation of quantum mechanics on path space, offered by Nelson's stochastic mechanics, is attractive to many. So far, it has been shown to be in complete agreement with the theory at the level of particle densities and spectra. But stochastic mechanics, because it operates at the sample-path level, seems to contain more information than the Schrodinger theory. The sample-paths in stochastic mechanics satisfy Nelson's generalization of Newton's equation (see §3 below); it is tempting, therefore, to conjecture that the sample-paths of the process provide us with the ensemble of actual particle paths. In this paper we explore one consequence of this conjecture. If the Nelson theory is to be than a very elegant reformulation of wave-mechanics, one must find areas where the predictions of stochastic mechanics and quantum theory differ. One area which merits investigation is the calculation of first hitting times in stochastic mechanics and first detection times in quantum theory. As a first step in this direction we consider the stochastic of the stationary states of the Hydrogen atom, a readily identifiable quantum system. In this paper we analyze in detail the diffusion process corresponding to the ground state of the Hydrogen atom; we obtain a skew-product formula for the process and give detailed results on first-hitting times. In [1] we describe how first arrival times of quantum particles can be calculated in the quantum theory of counting processes due to Davies [21. Finally, we indicate how our results may be extended to excited states. Since this is addressed to a mixed audience of physicists and

169

mathematicians, we have concentrated on explaining the ideas. The proofs are given in outline; details will be given in [1]. It is a pleasure to thank David Williams and Barry Simon for helpful conversations and to acknowledge SERC support through research grant GR/C113644.

§2. Diffusion Processes and Schrodinger's Equation

First we recall some results about diffusion processes (see [3] for details). Let X be a process on md satisfying the Ito equation (2•1)

dXt

where B is a BM(m d), a Brownian motion on md, so that each component Bj of B is a Gaussian process with and 1 , ... ,d.

(2.2)

Here E [ . ] denotes the expect at i on wi th respect to Wi ener measure P. If the drift b is sufficiently well-behaved, the process X has a transition density p(x,s ; y,t), defined for t>s by P[Xtf. A, Xs = x] = fA p(x,s ; y,t)dy,

( 2 .3)

and p is the fundamental solution of the forward Kolmogorov equation ap def -at = div y 0 gradyp - b f y t Jp ) c

(2.4)

subject to the condition p(x,t; y,t) =

s Lx-y

l.

(2.5)

The subscript y on the differential operators indicates that differentiation is with respect to the final point y. Regarded as a function of the starting point x, the transition density p satisfies =

+

b(x,t) gradxp

def

L*p,

(2.6)

the backward Kolmogorov equation. Here L * is the formal L 2 - adjoint of L. Now let f(x,t) be a classical solution of the equation

170

(2.7)

where V is a real-valued potential function. Multiplying by f*, the complex conjugate of f, and equating real parts, we find that the continuity equation ap + div j = 0 at

(2.8)

is satisfied by the quantum mechanical probability density p=lfl 2 and the quantum mechanical probability current j ;(f* grad f - f grad f*). Now assume that f is nowhere zero and write f exp(R + is), where Rand S are real-valued; then p = e 2R, j = e 2R grad S.

(2.9)

Hence we see that (2.8) may be written as

ap at

=divUgradr:

-pgrad

(R+S)).

( 2 . 10 )

Comparing this equation with (2.4), we recognize it as the forward Kolmogorov equation with drift b = grad (Re log f + 1m log f).

( 2 . 11 )

This leads to: Proposition 0 Let V be a real-valued potential function solution of the Schrodinger equation

..

f be a classical

( 2 . 12)

Su ose that f is nowhere zero and that the continuous function of (x,t) Ib(x,t)1


, and define the (7 )

If

L

aE T

of

SUI"

f

over

f

=(

A

fJ

and a hyperfinite set

by

f(a) J>._

is as in the e xar­p Le above and

g :R...,.R

is a function, then

according to this definition

L

t E T If

L

*cr(t).l­ -

N

ET

1

g(t )-N > n

n

n

1

is continuous, the sequence on the right converges to Jg(t)dt,

g

o

and thus

J1

o

g(t)dt

=

st

(

L

tET

1\

*g(t)Nj';

the Riemann integral is nothing but a hyperfinite sum. Let us take a look at Brownian motion. Fix an infinitely larae integer

N =( N >

n

and let

T

(T > n

be as in Example 1 ­ 10. The idea

188

is to obtain Brownian motion as the standard part of a random walk with infinitesimal steps. To be more precise, let w:T .... {-i,i}, and define a map

all internal maps X (w,t)

(8)

t

r

1

realvalued process b(w,t)

where

t

w(s)

this definition makes sense. To turn b: Dx [0,1] .... R, put

denotes the element in b

T

set of cardinality

peA)

X

into a

to the iwmediate left of

is a Brownian motion,

has internal cardinality

D

over a hyperfinite set

IN

I

2

N 1•

+

N+i,

t.

need a measure on w:T .... {-i,i}

This set consists of all internal functions

check that

by

st(x (to , t ) )

To claim that

Since

X:DxT .... *R

IN

s=O

{O'N'N"" ,t- N},

(9)

be the set of

i / N w(s)

Since I'm summing an internal function 1 2

D

D.

where

T

and hence it ought to be a hyperfinite

And so it is; in fact, it is trivial to

D (Dn>,where D is the set of all maps n is hyperfinite, I can for all internal A c D

w .... {-1,i}. n:Tn define

IAI

2 N+ i

This is a *R-valued, finitely additive measure on the algebra of internal subsets of

D. I can turn it into a R-valued, finitely

additive measure 0p

siwply by taking standard parts:

°P(A)

st(P(A)).

By Corollary 1 -9 the conditions of Caratheodory's extension theorem are trivially satisfied, and hence

0p

can be uniquely extended to a

measure on the a-algebra generated by the internal sets. The completion of this measure I'll denote by of

P. With the probability measure

Brownian motion. Although I

L(P) L(P)

and refer to as the on

D, the process

to sketch a quick proof of the a.s. with respect to the internal measure

Since

D

2-(N+i)

L

is a

continuity as this is the

only really nontrivial part. In the proof,

E(F)

b

shan't prove this in detail, I would like E

will denote expectation

P; i.e

F(w).

wED is hyperfinite, this makes sense for all internal functions

F.

The argument is based on the well-known expression

189

3(t-S)2 -

< 3(t-S)2

for the fourth moment of the random walk

X. This formula holds for

all finite random walks, ane by now you should be able to figure out why it also holds for the hyperfinite walk by

Q

m,n

Q

{w [ 3 i < 1113 sET n

m,n

To show that 0p(Q

X.

(m,n) EN 2, define a "bad" set

Here comes the proof: For each pair

m,n

b

is continuous, it suffices to prove that for each

as

) ... 0

n,

m"'co. Now

(10)

2}

.;; 2

n

'

where the last inequality cOmes from the following well-known ref leci i+1 I'i 1 . i+1 tion argument: If I X(w,-) -X(w,--) < t:» but there lS an sE i -] rn m n m i IX(w'm) -X(w,s) I1

1 > n' let

such that

I

Define

to be the reflected path

to

w(r)

I

4

w(r)

if

rs

Sw

be the smallest such

s,

w w

i -X(w,--) i+1 1 > -. 1 Slnce . X(w,-) there is an internal one-to-one m m n correspondence between reflected and unreflected paths, (10) follows.

then

4

I

The rest is just an easy computation. P(S'l

m, n

).;; 2

.;; 2

.;; 6n

m-1

,L

l=O

. . +1 l) p{wl [x(w'-m -x(w,lm )

m-1

L nE([x(w

. l

1

4 1

4

>

)

i=O

m-1

1.

i=O m2

6n ... 0 m

as

m'" O.

I have presented this proof in such detail because it illustrates one of the most interesting features of the nonstandard construction of Brownian motion;

namely, the way in which combinatorial arguments

(such as the reflection principle above) apply. In a sense,

b

has

the Donsker invariance principle built into it; it is at the same time

190

both a Brownian motion and a random walk. This possibility of doing combinatorics on infinite structures is one of the great assets of hyperfinite models in general. D. A fe1tl remarks on nonstandard methods. In order to "demystify" the subject, I have slightly misrepresented what nonstandard analysis is like. Just as nobody studies real analysis by considering equivalence classes of Cauchy sequences of rationals, nobody works in nonstandard analysis by constantly referring to equivalence classes of real sequences. If you take a closer look at what these sequences have been used for, you will realize that it has been to carryover operations and results from

R

*R. It turns out

to

that there is a theorem (aptly named "the transfer principle") which classifies once and for all which properties can be carried over in this way and with what consequences. Although it is not at all hard to prove, I have not presented this result here as it is phrased in terms of the socalled "first order languages" of mathematical logic which it takes a while to explain (see any of the books on nonstandard analysis listed in the references). Once you have the transfer principle, you can turn the situation around and use it to give an axiomatic description of what a nonstandard number system should be. Based on these axioms you can then develop the subject without any references to equivalence classes of sequences. I probably should point out that the axiom systems I'm talking about have other models than the structure A/*R constructed above (examples are appropriate quotient spaces R for sets

A

richer than

N), and that there thus is no unique set of

nonstandard real numbers. Although some people seem to find this lack of unicueness unsatisfactory for aesthetical reasons (and others use it as a pretext to turn philosophers [6]), it is not of any great practical consequence. As everybody knows, the construction of

R

from

Q

generalizes to

the construction of a completion for any given metric space. Is there a similar generalization of the construction of

*R

is "yes" ­ and very much so. In fact, given any set a nonstandard version

*M

from

R? The answer

M, you can obtain

by simply following the recipe in section A.

The reason why the nonstandard extension method is more general than the completion procedure, is, of course, that while in the latter case the equivalence relation on the space of sequences is defined in terms of the metric (and thus presupposes a metric structure), it's in the former defined solely in terms of the measure m on N (and thus doesn't require any kind of structure on

M). In the second part of

191

this paper, I ' l l make freauent use of the nonstandard extension

d

*R

Rd.

of

Nothing of what I have told you so far is new. A version of the socalled ultrapower construction (i.e. the "eauivalence classes of sequences" method that I have been explaining) was used already by Th. Skolem [13] in the first construction of a set of nonstandard naturalnmubers, and although A. Robinson often seemed to prefer to

*R

obtain

by a reference to the compactness theorem of first order

logic (see e.g. [12]), the systematic application of ultrapowers in nonstandard analysis was soon exploited by W.A.J. Luxemburg [11]. The hyperfinite model of Brownian motion and the underlying measure construction are more recent; they are due to R.M. Anderson [3J and P.A. Loeb [9J.

II. Perturbations of the Laplacian alono Brownian paths A. The problem and the results. The purpose of this second part of the paper is to give an impression of how nonstandard methods are applied. The results I am going to present are the outcome of joint work with S. Albeverio, J.E. Fenstad and R.

Hoegh­Krohn, and a more general and detailed account will aprear in

our book [2J. There is an alternative approach using Fourier analysis and ultraviolet cutoffs which has been developed by some of us in collaboration with W.

see [1J for a discussion.

I ' l l first give a description of the problem and the results in entirely standard terms. Assume that on a Hausdorff space E

x,

B

and that

be a symmetric, bilinear form on

is a completed Borel measure is a closed subset of X. Let 2 L and assume that E is

closed and bounded from below. II­1 Definition: A symmetric, bilinear form turbation of

E

supported on

(i)

F

(ii)

There is an

(iii)

If

B

F

is closed and bounded from below, and

f E D[EJ

E(f,g)

f E D[EJ

such that

D[F]

g E D[EJ.

D[E] .

F(f,f) *E(f,f).

vanishes in a neighbourhood of

for aile

is a per­

on

if

B, then

F(f,g)

192

I'm interested in the existence of perturbations supported on when

E

has measure zero. Although the

sets

B

and fairly general Markov forms

locally compact spaces situation where (11)

J

'2

R

d

on second countable,

Vf -9" dx

d,dx), L 2(R and

in

E

X, I'll restrict myself in this note to the

is the closure of

E 1

E (f ,g)

B

apply to all closed

B

is a Brownian path in

Rd.

If you think of the

Brownian path as a primitive model for a polymer molecule, you may consider a perturbation of

E

supported on

B

as the Hamiltonian form

of a quantum mechanical particle interacting with the molecule through a short range interaction. I'll say a few more words about the relationship to more sophisticated polymer models and

fields at the end

of the paper. There is an obvious strategy for constructing perturbations of supported on

B;

just let

bounded function on (12)

1f

F(f,g)

Rd

for all

p

E

A be a probability measure and a

and

B, respectively, and define VfVqdx + f'3:'fgdp B

fIg E

As

A and

p

vary, one would expect this

formula to produce a large class of perturbations of B. There is one problem, however; F

E

supported on

need not be lower bounded and

closable. On the other hand, even if all the F's are unclosable, there may still be perturbations of instance, that if E

B

E

supported on

has a perturbation supported on

B

if and only if

a perturbation can only be of the form (12) Since B

for some

B

if

d

d';; 3, but that

= 1•

is a d-dimensional Brownian path, i.e.

{b (to , t) E R w E Q,

d

I t E [0, 1 ] }

the natural measure

p

to use on

inherited from Lebesgue measure on [0,1]: p (A)

B; it's known, for

is a finite set (and not a Brownian path), then

= dx I

t. E [0, 1]

I b (to r t)

E A} •

I can now state the results I want to discuss:

B

is the one

193

Theorem: For almost all

w, the following holds:

(a)

If

d';;; 5, there is a perturbation of

(b)

If

d';;; 3, the form ( Lz ) is closable for all finite Borel functions

supported on

E

B

A.

When

d =4,5 , the perturbations of

E

will, in a certain sense,

be obtained from (12) by an infinitesimal choice of

A.

B. Hyperfinite quadratic forms. To prove the theorem above by nonstandard methods, I first have to rephrase the problem in nonstandard motion, let

X

1,X 2

be

, ••• ,

d

Beginning with the Brownian

independent copies of the hyperfinite

random walk defined in Part I, and put the size of the time increments, let

X

(X

6= 1

1,X 2

, •••

,X d)· If

and note that

X

1 N

is

is a

random walk on the lattice I ';;;6- 1 I'll leave it to you to check that If

66

A

for all U.

is a hyperfinite set.

is the discrete Laplacian on

A, i.e.

6- 2( L f(j) -2df(i»)\ 'lj-jJ=6 I, the form

where 1'1 is the maximum norm Ixl (13)

[(f,g)

--21

L

iEA

6,f(i)g(i)6d v

- defined for all internal functions

f

and

counterpart of (11). To get a of (12), let (14)

F(f,g} =

A:A-+*R

_2 L

2 iEA

g

- is a nonstandard

nonstandard representation

be an internal function and define 6

(i}g(i)6

d

+

L

tET

A(X(t})f(X(t})g(X(t»6t,

where The idea is to construct perturbations of turning the hyperfinite forms

F

E

supported on

E

by

into standard forms. Since the

standard and nonstandard forms operate on different spaces, it is necessary first to establish a correspondece between the elements of these two spaces. For each open cube

i= (n

I6,

... ,n

d6)

EA, let l i l be the half-

194

[i) If

f EL 2(Rd,dx)

of

f

, an internal function

f:A -+*R

is called a liftimr

if

I

iEA

6-

d

J

[i)

*f(x)dx - f(i)

d Obviously all f E L 2 (R ,dx) d 6- J *"f (x l dx ,

I

d 6 "" O.

have

La f

t Lnc s

j

just define

[i)

To turn

F

L 2(R

into a standard form OF on inf {st(F(f,f»)!f

where the values

±ro

d,dx)

f

by

f (i)

, let

0,

is a lifting of

just mean that

f

is not in the domain of OF.

The following (quite easy) result from chapter 5 of [2] shows that this definition is the right one. 11-3 Proposition. Assume that there is a F(f,f) ;;. zollfll2 F

for all internal

Zo ER

f, where

IIfli

such that =

(

I

iEA

)

Then

is a closed, symmetric, lower bounded form. Not very surprisingly, it turns out that ° E = E. Since I want to use

F

to construct a perturbation of

me when OF

and

E

E, I need a criterion which tells

are different. It will be convenient to have this

criterion expressed in terms of the resolvents of

E

and

F, and let

thus z)

-1

be the resolvent of

E, and G the resolvent of z result from chapter 5 of [2) now says.

11-4 Proposition. Assume that

Zo

Assume further that there exist a function

=

inf {st F (f ,f)

z E R,

F. Another easy

I

f

= 1}

is finite.

z < z 0' and an internal

f

with the following properties: E(f,f) and F(f,f) are both d finite, and (G z - Gz) f is a lifting of a nonzero function in L 2 (R ,dx). Then ° E o F.

*

The stage is now set for the proof of Theorem 11-2. Throughout this

195

w will be a fixed, "typical" element of

argument

rJ, where "typical"

means that it belongs to a set of measure one where all the claims I make hold. To compute the resolvent

I

F (f , g) = where

iEA

(-11. f( i) + A (i) v (i)

cd

u

(-11 + Av \

(x,y)

of

I {t I X (u.I,t) = L} I L'>t.

v (L) =

If

Gz

cd-

6

z)\

f ( L) )\ ( L) cd,

Thus

G (1 +

z\

F, note that

cd

G \-1

z)

denotes the kernel (or, if you like, matrix) of

I G (x,y)f(y)c yEA z then for all internal

G i.e. z'

d,

f

(15 )

(Xl ,y)

(-1) l

I

G (x,xl)A(x L )

(y,x1,···,xl)EA

Let

C

f (y) cd (L+1)

l+ 1 z

denote the "Brownian path" {X(w,t) It ET}; the expression above

can be written more compactly if we introduce the operators d) , L 2 (A, cd) ->L 2 (C,v) , : L 2 (c ,») ->L 2 (A,c G' : L 2 (C,V) ->L 2 (C,V) z Z defined by

a

G;:

I

yEC

Gz(x,y)f(y)v(y)

I G (x,y)f(y)C yEA z Gz'f(x) =

d

I G (x,y)f(y)v(y). yEC z

In this notation (15) becomes

196

\L AV Gz) 6d

Gz(-

(-1) L 8

z

z

z

and thus ( 16)

G

G I (-AG,)L, AG1\*z Z\L=O z) A

z

(

co

\

1\

Gz - G (1 +AG,)-1 A8* z z z

I have said nothing about the convergence of the infinite series involved in this calculation, but it is easy to check that the final answer is correct provided (++G') is a strictly negative operator (and (17)

I

(r1 + Gz) I

-1

z

'

thus exists). Since

' «X1 +Gz)g,g) L 2 (C,v)

+

[.,....-(1-:-) + I G (X,y)V(y)]C;(X) v Cx ) 1\ x yEC z

xEC

the operator 1 .,.----( )

z,,;;; zo' It is easy to check that it also implies

that

F(f,f)

Zo"f"2

for all

when

d > 3, in which case

f. Note that although the quantity

I G (x,y)v(y) always exists as an element in *R (it is the sum of an yEC z internal function over a hyperfinite set), it may be infinitely large since Gz("') has a singularity on the diagonal. This, in fact, happens A

must be negative and infinitesimal.

The only problem in showing that the standard part OF of F is a perturbation of we have chosen

E, is to prove that the two A

such that the function

h

are different. If above is negative, finite

and noninfinitesimal, it is not hard to check that the difference

197

(19 )

is a lifting of a nonzero function for a suitable choice of most "suitable" choice of

f

would have been

for the fact that Proposition 11-4 requires

a*-1 h Zo

f = E(f,f)

and

finite; as it is, one must work with approximations of

f

(the

if it weren't F(f,f)

8*-1 h Zo

to be

instead)

The difficulty now is that since it may happen that Zo =infCF(f,f) f o rrnu.La

II fll = 1 },

(19)

Zo

need is (19) with

I

is not sufficient for Proposition 11-4; what I replaced by a smaller

z

E R.

But

(20) and comparing (19) and (20), it is clear that all one needs to require z' -G'Zo is finite in some appropriate sense. A calculation similar to (17) shows that what is necessary is that the function is that

G

I

x'"

yEC

has finite when

(G (x,y) z

-G

L2

z ()

(x,y»v(y)

The last condition is obviouslv satjsfied

d"'; 3, because then

all negative

z

Gz(x,y)v(y)

L 2 (C,v)-norm for

has finite

E R.

'i G (x,y)v (y) is infinite for all real yEC z z, and at first glance it nay seem improbable that the differences When

d > 3, the quantity

(G z (x,y) - Gz (x,y»v (y) o can be finite. Observe, however, that by the resolvent equation (x,y) - G

Zo

where the sum over

(x,y)

(z-zo)

I

G

uEn z

(x .u )

d

(u,y) 0 ,

A smooths the singularity on the diagonal. In

fact, this smoothing is so efficient, that the function

I ( (x,y) -G (x,y»v(y) has finite L 2 - no r m when d"';S. We have yEC Zo thus shown that if d"';S, then OF is a perturbation of E supported

x

on

->

B

when the function

finite. Note that for

d

h

in

4,5

(18)

is

noninfinitesimal and

this implies that

infini tesinal. It is not hard check that for (12) is the standard part of representation of A.

F

when

A

A

is negative and

d"'; 3, the form

F

in

is a suitable nonstandard

198

There is an obvious way of reformulating the idea of this proof in standard terms; instead of working with a hyperfinite random walk on a hyperfinite lattice, I could have used finite walks on finite lattices and taken the limit as the mesh went to zero. That the hyperfinite coupling constant

A

is infinitesimal for

d

4

and

5, would in this

approach be reflected by the fact that, as the mesh got finer, I had to let my A'S converge

to zero to get convergence of the forms. This,

of course, is just an example of the philosophy I was propounding in Part I - that an infinitesimal is nothing but a convenient representation of a sequence converging to zero at a certain rate. The advantage of this representation is that you can do all your constructions and computations on one and the same structure, and that no limit has to be taken (compare the discussion of the nonstandard approach to Brownian motion at the end of Part I).

c.

A few speculative remarks on polymer measures and quantum fields. Formally, the (standard) forms we have constructed are given by

operators (21)

1

1

'2 6 + J A(b(s)

H

o(·-b(s)ds.

o

By the Feynman-Kac formula, the associated semigroup should be given by

t 1 JA(b(s)o(b(r)-b(s)dsdr

-J

Jf(b(t) )g(b(O)e 0 0

(22)

where

b

dP

is a new Brownian motion independent of

b. It is possible

to give precise meaning to and to prove this formula by using a nonstandard version of the Feynman-Kac formula (see [2J). Westwater [16J, [17] has shown that for

d

3

and

A

a positive constant, the exis-

tence of the exponential in (22) can be used to construct the Edwards polymer measures formally given by 1 1

-J JAo(b(r)-b(s)dsdr o0 e dW, where

W is Wiener measure. If we could extend Westwater's arguments

to the case where

d=4

and

A

is a negative infinitesimal, we would

get polymer measures in dimension 4. Through Symanzik's representation this question is intimately related to the nontriviality of and thus the arguments I have presented may be taken to indicate that the

199

"right" choice of the coupling constant in constructive

is

a suitable negative infinitesimal. This, however, is pure speculation; for what is known about nonstandard methods in quantum field theory, see chapter seven of [2J.

References 1.

S. Albeverio, J.E. Fenstad, R. Hoegh-Krohn, W. Karwowski, T. Lindstr¢m: In preparation.

2.

S. Albeverio, J.E. Fenstad, R. Hoegh-Krohn, T. Lindstr¢m: Nonstandard methods in stochastic analysis and mathematical physics, Academic Press, to appear.

3.

R.M. Anderson: A non-standard representation for Brownian motion and Ito integration, Israel J. Math. 25 (1976), 15-46.

4.

N.J. Cutland: Nonstandard measure Bull. London Math. Soc., 15 (1983),

5.

M. Davis: Applied nonstandard analysis, Wiley, New York, 1977.

6.

J.E. Fenstad: Is nonstandard analysis relevant for the philosophy of Synthese (to appear) .

7.

C.W. Henson, L.C. Moore: Nonstandard and the theory of Banach spaces, in A.E. Hurd (ed): analysis - recent developments, Lecture Notes in Mathematics 983, Springer, 1983, 27-112.

8.

H.J. Keisler: Foundations of infinitesimal calculus, Prindle, Weber and Schmidt, 1976.

9.

P.A. Loeb: Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Math. Soc., 211 (1975),113-122.

and its applications, 89.

10.

R. Lutz, M. Goze: Nonstandard analysis, Lecture Notes in Mathematics 881, Springer, Berlin - Heidelberg - New York, 1981.

11.

W.A.J. Luxemburg: Nonstandard analysis, Lecture notes, Pasadena, 1962.

12.

A. Robinson: Non-standard analysis, North-Holland, Amsterdam, 1966.

13.

Th. Skolem: Uber die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder unendlich vieler mit aussliesslich Zahlenvariablen, Fundamenta Mathematicae, (1934), 150-161.

14.

K.D. Stroyan, W.A.J. Luxemburg: Introduction to the theory of infinitesimals, Academic Press, New York, 1976.

15.

K.D. Stroyan, J.M. Bayod: Foundations of infinitesimal stochastic analysis, North-Holland (to appear) .

200

16.

J. Westwater: On Edwards' model for long polymer chains, Comm. Math. Phys., 72 (1980), 131-174.

17.

J. Westwater: On Edwards' model for polymer chains III, Comm. Math. Phys., 84 (1982), 459-470.

HAUSSDORF DIMENSION FOR THE STATISTICAL EQUILIBRIUM OF STOCHASTICS FLOWS

1. Let M be a compact manifold of dimension d, and (Tn,n a family of i.i.d random CZ-diffeomorphisms of M. For all m,n E m < n, we note the comT posed transformation T ... Tm. n_1 n_Z Let Q be the associated markovian transition probability kernel: Qf(x)

E(f(Tnx»

forall

»c

z,

XEM

and ff'C(M).

We shall assume in the following that Q is ergodic and denote its invariant probability measure by A(dx). Then, for any f E C(M), n F PF

is a bounded martingale in

p and it converges a.s. towards a limit

is a random probability measure on M. We call it the statistical equilibrium of the flow S at time n. (cf our previous works [4J, [5J, [6J , [7J, where it was defined in the context of continuous flows defined by S.D.E's). The law of is clearly independent of

nand

Pm for m < n.

Concrete examples on the torus were studied in ([4J,[5J) where the statistical librium happened to be a.s diffuse for d e 3 and a.s. Dirac measure for d 1 or 2. In [OJ, Baxendale studies a flow on the sphere where the statistical equilibrium appears to be a.s. a Dirac measure. In [6J, [7J the exemple of isotropic flows on the f ] at space is studied. (In that case, the invariant measure, which is Lebesgue measure, is infinite, and other problems arise). The purpose of this work is to give a majoration of the average Haussdorf dimension of P in terms of the Lyapunov spectrum of the flow. The method is a rather straightforward extension of the argument given by Ledrappier in [2J for classical dynamical systems (cf also references in [2J). See [1J for other applications of methods of classical dynamical systems to the context of stochastic flows.

(*) UNIVERSITE PARIS VI - Laboratoire de Probabilites - 4 place Jussieu - Tour 56 3eme Etage - 75230 PARIS 05

202

We assume that

E(sup n ;

n)

are obviously

203

2. To give a precise statement, we need a few definitions and comments. a) For any positive random probability measure = lim

0+0

lim sup E(Log

t

on M, we set

/ Log(l)

s + 0

s

we need to where is the smallest number of open balls of radius cover M up to a set of strictly smaller than o. Note that we can assume that the balls are centered on a countable dense subset of M without changing which is therefore a measurable function of

u,

c s-d for some constant c. Let (Q,l{,P) be the proNote also that bability space on which the Tn are defined. One can assume that there is an ergodic shift

e such that

Set Q+

o(Tn,n

0)

an ergodic shift on (Q

Tn+ 1(w)

Tn(ew).

and CN-

o(Tn,n

x




u(

/

/ II u II '" (1

+ x)

n

e

for all

E(,J (AUJ,x,n)) increases to one as n t "', o w,x,n 2 2 w,x,n 2 1- n => (A ) ;;: 1-n such that: )) o =

than

of probability less

11.

?;

Set For n

for all

-s-

j

n1( n ) v n2 ( n , x ) , and

are satisfied by Therefore, for any 8

!(I

=

11

inf k,

0,

n

L

1

A·

1

0


Vv>ga.e.

w

Cj

(1.1 )

a.e.

where (f,E) is a regular Dirichlet space, g is a given function belonging to f, v is a difference of positive Radon measures of finite energy integrals, a > 0 and H(x,u)

- sup {f(x,z) - c(x,z)u} z

f(.,z), C(.,z) E L2 . Inequality (1.1) corresponds to a stopping problem for the symmetric Markov process associated with (f,E) , where killing rates and cost functionals represented by f and c are controlled besides stopping times. treat the characterization problem in § 3 after considering the existence and uniqueness of the solution of the inequality (1.1) in § 2. § 3 is a generalization of [6] and similar problems have been considered in [3], [5J and [9J. Inequality (1.1), when specialized to the diffusion case with regular coefficients, is reduced to a special form of Hamiltonhowever treat general symmetric Jacobi-Bellman inequation (cf. [3J, [5], [9]). Markov processes including jump type processes and large classes of obstacles. By employing potential theory of Dirichlpt space and Markov processes developed in [4], we establish the relationship between the variational inequality (1.1) and a stopping problem without regularity arguments on the solution of (1.1). In §4 we treat a problem in passage to the limit in the case where obstacles converge monotonely. We note that Theorem 4.1, combined with Theorem 3.3, indicates that a pay-off function of our stopping problem can be characterized as a quasicontinuous modification of the solution of the variational inequality even if the obstacle is far from a regular function.

209 § 5 treats another problem in passage to the limit in the case where the obstacles gn converge to a function g in the sense of capacitary integrals.

The results in § 4 and § 5 are generalizations of [1], [2] and

[7].

The full proofs of the results will appear elsewhere ([8]). here we shall limit ourselves to comment on some main ideas or sketches of the proofs. 2 ­ VARIATIONAL PROBLEMS IN TERMS OF DIRICHLET SPACE Let m(dx) be a non­negative everywhere dense Radon measure on a locally compact Hausdorff space X with a countable base. We denote by (F.E) a regular Dirichlet space relative to L2(dm). Let f be a separable metric space and f(x.z) and c(x.z) be functions on X x f satisfying: (i) (ii)

f(x •. ) and c(x •. ) are continuous for m­a.e.x c( .• z) is measurable for each z and there exists M such that c(x.z) M for each z and m­a.e.x f( .• z) E L2(m) and there exists fo(x) E L2(m) such that If(x.z) I ;'; fo(x) m-a ,e . for each z.

o< (iii)

We then define a function H(X.u)

=

H(x.u)

on X

x

IR

1

by

­sup {f(x.z) ­ c(x.z)u} zEf

(2.1 )

We note that H(x •. ) is a non­decreasing function and that H( .• u(.)) E L2 (m) if u(.) E L2(m). Therefore. if we define an operator H by Hu

H( .• u( . )).

then H is a map from L2(m) (Hu 1

u E L2(m) , to

L2(m)

and has a monotone property:

2 Hu 2• u1 ­ u2) ;; 0, u1.u 2 E L (m)

For given positive Radon measures VI and v 2 of finite energy integrals, we first consider the following equation: (2.2)

where v = VI ­ v 2 and Ii denotes a quasi­continuous modification of

vEF •

210

Next, we consider for given function

g E F the following variational inequal-

ity + (Hw,v-w)

>

(2.3)

where Kg

=

{w

E F;

v

g m-a.e.} .

Then we have the following propositions. Proposition 2.1: For any positive Radom measure vI grals, (?.2) has a unique solution. Proposition 2.2:

and v2 of finite energy inte-

For each g E F, (2.3) has a unique solution.

The proof of Proposition 2.1 consists in showing that Banach contraction principle applies to the operator Tw = Ua+Mv - Ga+M(Hw Mw) . For the proof of Proposition 2.2 we employ an approximation method. We consider inductively the following sequence of variational inequalities:

Ea", n

n

,v-u n ) -

-

(Hu n_1 - Mu n_1 , v-un)

= 1,2, ...

We can show that the sequence {un} of the solutions of the above variational inequa1 iti es converges to some u E F, whi ch is the solution of (2.3). Uni queness follows from monotonicity of the operator H . 3 - STOCHASTIC CONTROL OF SYMMETRIC MARKOV PROCESSES

Let M = (D,B,Bt,Px'X t) be a symmetric Markov process on X and we assume that (F'E) is regular. Let f(x,z) and C(x,z) be functions satisfying (i), (ii), (iii) in § 2. We moreover assume that f(. .z ) and C(. ,z) are quasi-Borel functions. Let v be as in § 2 a difference of two posi t i ve Radon measures vI and v2 of finite energy integrals and At be a continuous additive functional of M corresponding to v. We consider the following stochastic control problems:

*

u (x)

t

=

rr""-at-!c(Xs,Zs)ds sup Ex I. J e 0 f(Xt,Zt)dt 0 {Zt }EM r

(3.1)

211

t

r '-at - J0 C(X s ,Zs )ds dA + J e t 0>

o

T

w*(X)

Joe

-at -

t

J

(3.2)

0

where g is a given quasi-continuous function belonging to {Zt} is a progressively measurable r-valued process}.

F and

Mr

{{Zt};

Then we have the following Theorems.

- by (3.1) is a quasi-continuous modification of the solTheorem 3.1: u* (x) deflned ution u of the equation (2.2). Theorem 3.2: w*(x) defined by (3.2) is a quasi-continuous modification of the solution w of the variational inequality (2.3). The crucial point of the proof of Theorem 3.1 is the following: Let u be the solution of (2.2), then t

e

-at- 10 C(\,Zs)ds t

+

J e-

as

s

1OC(X,Z )dT TT

o

dA

s

is a martingale for each {Zt} E Mr, where t

As

As -

J o

Regarding the proof of Theorem 3.2, we note that the solution w of (2.3) can be written

for some positive Radom measure w of finite energy integral and that a continuous additive functional of (Px ,X t) corresponds to the Radon measure w. Then we can show that

212

e

-ct - ftC(Xs'z )ds o s w(X ) + t

Jt e -as-f 0

C(X ,z )dT T

T

o s

f C(X ,z )dT OTT

is a martingale, where At

dA

s

t

=

]J At + At - Jr H(Xs,w(Xs))ds .

o

Further observation concerning the support of the measure of Theorem 3.2.

]J

leads us to the proof

Remark 3.1: Let us put Q = {g: g is a quasi-Borel function such that there exists u E F satisfying -u 9 q.e.} . If for

9 E

Kg

Q we set a closed convex subset Kg of {v E F;

v

9

F

as

q.e.} ,

then we can see in the same way as in Proposition 2.2 that the following variational inequality (3.3) has a unique solution.

(3.3)

Replacing (2.3) by (3.3) we obtain the following version of Theorem 3.2. Theorem 3.3: If 9 is a quasi upper semi-continuous function and g E Q, then the same statement as Theorem 3.2 holds by replacing (2.3) by (3.3). In the same way as the proof of Theorem 3.2 the solution w of (3.3) can be written

for some positive Radon measure. Concerning the arguments on the support of the measure ]J we use the fact that we can take a non-increasing sequence of quasi-continuous functions which converges quasi-everywhere to g.

213

4 -

PASSAGE TO THE LIMIT (I)

We consider a problem in passage to the limit in the above stochastic control problem. For a given sequence gn of functions belonging to Q we put

[ Jr

T

Ex

o

e

-ct - ftC(X ,Z )ds s

0

s

f(Xt,Zt)dt (4.1 )

T

+

Je

-at- ft C(X 0

)ds

s

dA

o

t

+ e

-aT -

f T C(Xs'Z )ds 0

s

] g (X ) . n

T

Corresponding variational inequality to (4.1) is the following:

(4.2)

By Remark 3.1 we can define an operator from Q to F such that = wn ' 9n E Q where wn is the solution of (4.2). Then we have the following theorem. Theorem 4.J.: Let {g} be a non-decreasing sequence converging quasi-everywhere to a function 9 E Q nand assume that w* . . . . . . q.e., then w*n converges quasi* n n everywhere to a function w such that

w*(x)

e

-o t - ft C(X ,Z )ds

s

0

s

(4.3) +

*

Moreover w

Io T

-at - ft C( Xs ,Zs )ds -nr e 0 dA + e t

fT 0

C( X ,Z )ds SSg( \

] )

•

....., w(g)

The proof of the present theorem is based on the fact that a non-decreasing sequence of a-potentia-Is whose energy integrals are boundedly dominated converges strongly to some a-potential.

5 - PASSAGE TO THE LIMIT (II) We introduce a function space L(C)

L(C)

by

{c : ¢ is a quasi-continuous function such that there exists u E F satisfying u > I ¢ I q.e.} .

214

And we define

and D(¢)

C(dl)

C(¢)

inf

IJ(¢)

I

for

{Ea(U,U) ;

cap {I ¢ I

>

L(C)

as follows

u E F, u

>

I ¢ I q.e.}

t} dt 2 .

0

Here "cap" means capacity defined in terms of Di richlet form Ea' We remark that L(C) is a Banach space with norm VC(¢). Denoting by To the restriction of the operator T to L(C), we can state continuity theorem concerning this map To' We set Ilvll = VEa(V,V). have the following theorem and Corollaries. Theorem 5.1:

For gl

and g2 belonging to

L(C)

we have

Here K1,K2 and K3 are positive constants independent of

gl

and g2'

Corollary 5.1: Let be a pay-off function defined by (4.1) for given gn E L(C). For each n , if gn --> 9 in L(C) as n -->00, then 0 as n -->00, where w* is a pay-off function defined by (4.3) corresponding to g. Corollary 5.2: we have

Let gn,q E L(C) and assume that as n -->00.

D(gn- g) --> 0 as

n -->00, then

Let us put wI = To(gl) and w2 = To(g2)' then there exist positive Radon measures VI and u2 of finite energy integrals such that

By using monotone property of

H we obtain

VE (U u. ,U u·) a a 1 a 1

is dominated by for each i, then we have Theorem 5.1.

Corollary 5.1 is a direct consequence of Theorem 3.2 and Theorem 5.1. For the proof of Corollary 5.2 we observe that C(¢) 4D(¢) .

215

The author wishes to express his hearty thanks to Professor Dr. S. Albeverio who invited him to the 1st BiBoS symposium and gave him valuable advice to publish this article.

REFERENCES [lJ

D.R. Adams: Capacity and the obstacle problem, Appl. Math. Optim. 8 (1981) 39-57. -

[2J

H. Attouch, C. Picard: Inequation variationelle avec obstacles et espaces fonctionnels en theorie du potentiel, Applicable Anal. (1981) 287-306.

[3J

A. Bensoussan, A. Friedman: Nonlinear variational inequalities and differential games with stopping times, J. of Funct. Anal. (1974) 305-352.

[4J

M. Fukushima: Dirichlet forms and Markov processes, North-Holland/Kodansha (1980).

[5J

N.V. Krylov: Control of a solution of a stochastic integral equation, Theor. (1972) 114-131. of Prob. and its Appl.

[6J

H. Nagai: On an optimal stopping problem and a variational inequality, J. of Soc. of Japan 30 (1978) 303-312.

[7J

H.

[8J

H. Nagai: Stochastic control of symmetric Markov processes and non-linear variational inequalities, to appear.

[9J

M. Nisio: On non-linear semi-group for Markov processes associated with optimal stopping, App. Math. and Optimization 4 (1978) 143-169.

Impulsive control of symmetric Markov processes and quasi-variational inequalities, Osaka J. of Math. (1983) 863-879.

Author's Address: Hideo NAGAI INRIA Domaine de Voluceau-Rocquencourt BP. 105 F-78153 LE CHESNAY Cedex / France Department of Mathematics Tokyo Metropolitan University FUKASAWA, Setagaya, Tokyo Japan

MEAN EXIT TIMES AND HITTING PROBABILITIES OF BROWNIAN MOTION IN GEODESIC BALLS AND TUBULAR NEIGHBORHOODS Mark A. Pinsky Mathematics Department Northwestern University Evanston, IL 60201 In this lecture I will first survey some recent results on Brownian motion in small geodesic spheres of a Riemannian manifold. Then I will turn to some corresponding questions in tubular neighborhoods of a submanifold. 1.

Diffusion processes on manifolds.

We first review the modern formulation of diffusion processes. Let M be a differentiable manifold of dimension n and let A be a second order differential operator on M. A diffusion process X = (\,P x) is said to be generated by A if for every twice differentiable bounded real­valued function f with bounded derivatives t is a P martingale. x

t

f(\) ­ J (Af)(Xs)ds

+

o

This has the following consequences:

(1.1 )

A is positive semi­definite

(1.2)

E f(Xt) = f(x) + E J (Af)(X )ds x x 0 s

t

For every bounded stopping time

(1.3)

T

(t

> 0)

T

E f(XT) = f(x) + E J (Af)(Xs)ds x x 0

The latter is Dynkin's identity [D] which can be thought of as a form of the fundamental theorem of calculus for strong Markov processes. In this direction, we can further replace the integral term by higher powers of the generator and obtain the "stochastic Taylor formula" [AF,AK] written as follows: (1.4)

N

f(x) ­ E f(X ) = L k=1 x T

k

..l=.!.L E nkAkf(X )} + k!

x

T

N+l

(­1),

N.

T

E J sN AN+lf(X )ds x 0 s

The function f is supposed to be infinitely differentiable with bounded derivatives. Dynkin's identity (1.3) immediately yields the probabilistic representation of two classical boundary value problems: (1.5)

Af 1

­1

(1.6)

Af

0 in

2

in

B, B,

f

1 = 0 on

aB

f 1(x)

Ex (T)

f 2 = 0: Xt

f

B} •

217

The stochastic Taylor formula (1.4) gives the following probabilistic representation for the biharmonic equation (1. 7) Higher order equations are solved similarly. In practice one can rarely find an exact solution of these classical equations. Nevertheless Dynkin's formula yields immediately the following representation of approximate solutions: (1.8) diam B£ + 0 The observation Here B£ is a sequence of open sets with x E B£ (1.8) will be systematically exploited to study Brownian motion in small spheres.

n Let (M ,g) be an n­dimensional Riemannian manifold and bE x(M) field. The differential operator

I

a smooth vector

generates a diffusion process, where is the Laplace­Beltrami operator of the metric g. We note that any second­order strictly elliptic operator may be written in this form for a suitable choice of (b,g) Let T be the exit time from a ball of radius £ with center at mE M £

\

=

inf {t > 0: d(Xt,m)

=

e}

Here d is the distance function determined by the Riemannian metric. Our first result gives the joint influence of (g,b) on the mean exit time [MP 3]. Theorem O.

For each

m

M we have for

£ + 0

where c O,c1 are dimension constants and T is the scalar curvature of the Riemannian metric. This is proved by finding a function f satisfying Af = ­1 + 0(£4) and then £ £ applying Dynkin's formula (1.3); f£ may be found, for example, by solving an ordin­ ary differential equation in the radial variable and averaging over the unit sphere. In order to discuss inverse problems in stochastic Riemannian geometry, one must obtain an additional term in the expansion of Theorem O. Indeed, knowledge of the scalar curvature only suffices to determine the Riemannian metric in dimension two! To obtain the additional term we restrict attention to the case b = 0 , for technical reasons only. The following result was proved in [GP].

218

Theorem 1. Riemannian manifold

T ,g)

the exi t t i of the diffusion generated by 6. on a Then there exi..?! ci c i (n) 0 < i < 5 such that

.. ) is the Ri cci tensor computed where R = (R i ajb) .!.2. the Ri emann .!-ensor and_ o = (p lJ -------any orthonormal bas is the space Mm Furthermore c.1 > 0 for i * 3 c2 + c 3 = 0 •

The inverse problem is formulated as follows: Conjecture. If for each mE M, Em(T E) = COE 2 then (Mn,g) is locally n metric to (R ,gO) • We have obtained a positive result in low dimensions, as follows: 2 8) Corollary 1. Let 2 in < 6. If for each mE M, Em(T E) = CoE + 0(E then n (M ,g) .!.2. locally isometric to (R ,gO) • The algebraic mechanism behind Corollary 1 is apparent when we write the quadratic part of the curvature term as follows for n > 2 : n

2 c 2 IR 12 + c 3 1p 1

=

n:-z

2 2 c 2 ( 1W1 + 6 - n 1P 1 )

where W is the Weyl conformal curvature tensor. The above quadratic form is positivedefinite if n < 6. In fact for n = 6 there is a non-trivial null space which is concretely presented as follows: product of sphere of constant sectional curvExample. Let M6 = 53 x H3 , ature +1 with a hyperbolic space of constant sectional curvature = -1. Then 2---10

---

-

Em(T E) COE + O(E r • This is the simplest of a family of Riemannian symmetric spaces with the indicated asymptotic form of the mean exit time. The other members of this family are all of the form M = G x GC where G is a compact Lie group with the bi-invariant metric and GC is the non-compact dual obtained by complexification. The smallest dimension for which G is non-flat is n = 3, hence the above example. In order to obtain a positive result in higher dimensions, we consider the second moment Em(T E) for which we have obtained the following asymptotic formula [MP 2J. Theorem 2. such that

Under the hypotheses of Theorem 1, there exi st_ di

d (n)

i

,

0

0). In [2] it was shown that (1.1) has exactly one solution depending vaguely continuously on the parameter s. For s=O the solution is explicitly obtainable, namely (oa=Dirac measure supported by a). Hence, crudely speaking, as s approaches zero, the entire probability collapses into the origin. Intuitively, one expects that an s-dependent rescaling of the abscissa may lead to a non-trivial limit as s tends to zero, providing thus a more detailed description for the small s behaviour of

and quantities derived thereof. This scaling approach based on intuition,

insight and (numerical) evidence has been remarkably successful [3-6] but remains "without mathematical justification" [7]. A rigorous formulation is presented in Section 2. The proof is split into two parts contained in Sections 3 and 4.

225 2. The Scaling Laws Let M be the set of finite measures with support in R+ satisfying

(2.1) 1

I

I

The subset M c M is characterized by equality in (2.1); the subset M c M is o

characterized by

(2.2)

A({O}) = 0,

We set (JA) (x)

A([O,X))

°

(2.3)

(y,z)

(2.4)

>

x

for A E M; it determines A uniquely. For p £

and s (JR

s

° we define Rs by

A)(X)

ffdp(y)dA(z)X

S,x

with X being the characteristic function of the set C defined in (1.2). s,X I 1 S,x 1 Obviously, A E M implies RSA EM; hence, R will be considered as a map from M s into itself. In [2] it was shown that R has a unique fixed point, E MI. Actually,

E

1 M for s > a

tinuously on s; as

s

° and I

s

s

depends vaguely con0 a s E M , this implies also weak continuity [8]. Rescaling is

now introduced as follows: Let

= 0 . Furthermore,

e

be a continuous increasing function defined on

(O,a), a > 0, satisfying lim

O(t)/t

°

(2.4)

Define the rescaled probability measure

o

associated with

o It is the unique fixed point of




N. There exists a continuous function u with compact support in R+ satisfying 0
0 arbitrary. Differentiate f(s) == exp(-u(s»

u(s)

x] (1+sx)

(3.10)

twice to obtain flIes)

g(u(s»

g(u)

4 3 exp(-u)(u -2u )

(3.11)

By an elementary calculation

I g(u) I

< g(3+..j3)

2C

C

6 (l2+7..j3)e -3-..j3

(3.12)

Hence,

I s-l(f(s)

- f(O) -

Cs

(3.13)

229

A change of variables and f'(O) = e-xx Z lead to (3.14)

which implies Lemma Z. Lemma 3. Let p transform

p

£ N k=I,Z,3. The asymptotic behaviour for p+O of the Laplace k, is given by

k = 1

cp I-p(p) '"

cp I.£n cp

I

(3.15 )

k = Z

(cp ) I-a rea)

k

=3

Proof. I,

Use p-l(l_e- Px)

lim p-l (l-e- Px) p+O

integrate with respect to p

£

(3.16)

x

N, and apply Lebesgue dominated convergence

theorem. Z.

Note that P

-1

(I-p(p»

G(p)

fG(x)e- px dx

(3.17)

with G defined in (Z.II). Set G (x) = min(l,c/x) and define p via (Z.11); obviously, p £ N As the Z' 0 0 0 1, difference G-G is in L it is sufficient to prove G (p) '" cl.£n cpl which o

0

follows from Go(p) = (l-exp(-cp»/p + c E ( cp ) and the asymptotic behaviour 1 of the exponential integral E [IZ]. 1 3.

Application of Theorem 1, p.456 [13] yields G(p) '" r(a)c

I-a

, hence (3.15).

Corollary. k

= I,Z

k = 3

(3.18)

230 Now, take a c-sequence with limit

From (3.3), (3.4), Lemmata 1 and 2, and

Corollary we obtain 2 d Pcp) dp2

pep) f(Ci)p

-Ci

Pcp)

k

1,2

k

3

(3.19)

leading to A exp(-p) Pcp) ==

with

°< A < 1

(3.20 )

A p(Ci)(p)

(possibly depending on the c-sequence) where

is the proba-

bility measure with density f(Ci) given in (2.16) (solving (3.19), k==3, yields an expression containing a modified Bessel function

a Mellin transformation then

leads to (2.16), details in [11]). From (3.20) we obtain A °1 A

(ci)

k

1,2

k

3

(3.21)

in particular,

°

(3.22)

Lemma 4. The value of A in (3.21) is A==l independently of the choice of the c-sequence. Proof. In Section 4. Lemma 5. Let X be a sequentially compact Hausdorff space, a >

° and

f

o: 0, (F (t )) convergent) have the same non limit, say x o' then F: [O,a) X with F(t) == Fo(t), t > 0, and F(O) == X is o x continuous at t==O; in particular, t 0 implies F(t n n) o' (O,a)

X. If all c-sequences (i.e. t

Proof. By contradiction. As M is Hausdorff and vaguely compact we may apply Lemma 5 to F : (O,a) Fo(t) ==

o

M with

As here F is continuous the same is true for its extension F. This o completes the proof of the theorem.

231

4. Proof - Part II One verifies easily [2] that (JBX)(x) = A((x 1

-1

,00»

(4.1)

1

1

defines a map B: M M As p and belong to M we may define t=Bp and at o o o The latter is the fixed point of T where t

,x with

,x

(4.2)

(y,z)

being the characteristic function of E

(B(t) + 1 )-1 < x } ty z+t

t,x

(4.3)

Applying the Stieltjes transformation (SA)(q) = f(x+q)

-1

dA(x)

q

> 0

(4.4)

to the fixed point equation for 0t yields

o

(4.5)

with

(4.6)

. f Lemma 6. For q

(St j

dt( ) ( + B(t) q(X+t»-l y y t x+q+t

0 the asymptotic behavior of St is given by

Cq ) '"

c

k

1

clQn cql

k

2

k

3

c1-Ci

-Ci

q

(4.7)

Proof. 1.

1 N satisfies fdt(x)x- = c < 00. Hence, the result follows from 1 Lebesgue dominated convergence theorem. t=Bp, P

£

232

Z.

A partial integration yields (L({O}) (SL)(q)

f(x+q)

0)

-Z (JL)(x)dx

(4.8)

For L=6p, P £-N 1

L (x

-Z

the difference JL-JL with (JL )(x) = min(cx,l) is in Z 0 0 ). Hence, the result follows from c Qn 1+cq cq

(SL )(q) o

3.

(4.9)

An elementary calculation yields (4.10)

(SL)(q)

From (3.15), k=3, and Theorem 1, p.473 [13J we obtain (SL)(q)

f(a)f(Z-a)c

I-a q -a

(4.11)

which is (4.7), k=3. Let C be the set of continuous functions on R+ with a finite value at infinity f (i.e. f e C iff there exists c e R such that f-c e Co), f The functions Kt(o,q) defined in (4.6) belong to C and converge in supremum norm f as to Ko(o,q) e C where f (x+q)-Z (I-x Z) (x+ )-Z{l _ q

s a.nrto

xz-a

k

= 1,Z

k

=3

(4.1Z)

as is seen using Lemma 6. 1 Lemma 7. Let (A ) be a sequence in M converging vaguely to A e MI. Let (f ) be a n

sequence in C converging in supremum norm to f f

lim

f

f

n

axn

£

C Then f.

f fdA

Proof. By "shifting" f n and f to Co and using An(R+) reduced to Lemma 1.

n

(4.13)

1 the problem is

233 For any c-sequence

converges vaguely to

given by (3.21). Hence

con-

verges vaguely to

a = (I-A)6

o

+ A •

k

1,2

k

3

(4.14)

due to (3.22) a is nondefective. Application of Lemma 7 yields

f

K (x,q)da(x) o

o

(4.15)

From (4.12) we obtain = q

-2

(4.16)

The equations (4.14) - (4.16) are compatible only for A=I, q.e.d ..

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

Dyson, F.J.: The dynamics of a disordered linear chain. Phys.Rev. 92, 1331-1338 (1953). Schneider, W.R.: Existence and uniqueness for random one-dimensional lattice systems. Commun.Math.Phys. 87, 303-313 (1982). Bernasconi, J., Schneider, W.R., Wyss, W.: Diffusion and hopping conductivity in disordered one-dimensional lattice systems. Z.Phys. B37, 175-184 (1980). Alexander, S., Bernasconi, J., Schneider, W.R., Excitation dynamics in random one-dimensional systems, Rev.Mod.Phys. 53, 175-198 (1981). Schneider, W.R., Bernasconi, J.: In: Lecture Notes in Physics, Vol. 153, pp.389-393. Berlin, Heidelberg, New York: Springer 1982. Nieuwenhuizen, Th.M., Ernst, M.H.: Transport and spectral properties of strongly disordered chains, Phys.Rev. B31, 3518-3533 (1985). Kawazu, K., Kesten, H.: On birth and death processes in symmetric random environment, J.Statist.Phys. 37, 561-576 (1984). Bauer, H.: Wahrscheinlichkeitstheorie und Grundzuge der Masstheorie. Berlin, New York: Walter de Gruyter 1974. Raina, R.K., Koul, C.L.: On Wey1 fractional calculus. Proc.Amer.Math.Soc. 73, 188-192 (1979). Braaksma, B.L.J.: Asymptotic expansions and analytic continuations for a class of Barnes-integrals. Compos.Math. 15, 239-341 (1963). Schneider, W.R.: Stable distributions are-Fox functions. Preprint (1985). Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. New York: Dover 1965. Doetsch, G.: Handbuch der Laplace-Transformation. Basel, Stuttgart: Birkhauser 1971.

A RIGOROUS APPROACH

ASYMPTOTIC FREEDOM

R. SENEOR Centre de Physique Theorique de l'Ecole Poly technique Plateau de Palaiseau - 91128 Palaiseau - Cedex - France

I

INTRODUCTION In the last 10 years a lot of progress has been made in the rigorous cons-

truction of field theories. If the results obtained up to last year had to deal essentially with "academic models", i. e. scalar models in low dimension, there was in 84 a great progress made when people realized that asymptotically free theories are roughly similar to superrenormalizable field theories the main difference being as we shall see later in the slower rate of convergence of the former ones. I will report on this progress. Up to now it is far from giving us the possibility of controlling* "physical models", I mean by that the non abelien gauge theories (which are asymptotically free), but clearly are did half of the way if we add the fact that T. Balaban at Harvard has developped a program for the control of superrenormalizable gauge theories (i.e. 2 and 3 dimensional gauge theories). The work I will describe is the result of a collaboration with J. Feldman, J. Magnen, V. Rivasseau ([1]). Similar results obtained by a different method have been given by K. Gawedzki and A. Kupianen ([2]). The idea of the proof is due to the encounter of 2 different methods the phase space cell expansions which originated in J. Glimm and A. Jaffe ([3]) and where extended to the study of superrenormalizable massIes field theories ([4]) where they have to be combined with scaled cluster expansions the resumrnation of the most divergent parts of the coupling constant counterterms used by G. 't Hooft ([5]) and V. Rivasseau ([6]) for the construction of planar

2 , described by a Gaussian mea-

of covariance

c

­m Ix­yl o

e

C(x­y)

d-2

Ix­y I

(II. j)

and mean zero. Roughly speaking I

e I

­2:

fe-2:

dv c (