Functional Integration and Partial Differential Equations. (AM-109), Volume 109 9781400881598

Table of contents :
CONTENTS
PREFACE
INTRODUCTION
I. STOCHASTIC DIFFERENTIAL EQUATIONS AND RELATED TOPICS
§1.1 Preliminaries
§1.2 The Wiener measure
§1.3 Stochastic differential equations
§1.4 Markov processes and semi-groups of operators
§1.5 Measures in the space of continuous functions corresponding to diffusion processes
§1.6 Diffusion processes with reflection
§1.7 Limit theorems. Action functional
II. REPRESENTATION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS AS FUNCTIONAL INTEGRALS AND THE STATEMENT OF BOUNDARY VALUE PROBLEMS
§2.1 The Feynman-Kac formula for the solution of Cauchy’s problem
§2.2 Probabilistic representation of the solution of Dirichlet’s problem
§2.3 On the correct statement of Dirichlet’s problem
§2.4 Dirichlet’s problem in unbounded domain
§2.5 Probabilistic representation of solutions of boundary problems with reflection conditions
III.BOUNDARY VALUE PROBLEMS FOR EQUATIONS WITH NON-NEGATIVE CHARACTERISTIC FORM
§3.1 On peculiarities in the statement of boundary value problems for degenerate equations
§3.2 On factorization of non-negative definite matrices
§3.3 The exit of process from domain
§3.4 Classification of boundary points
§3.5 First boundary value problem. Existence and uniqueness theorems for generalized solutions
§3.6 The Hölder continuity of generalized solutions. Existence conditions for derivatives
§3.7 Second boundary value problem
IV. SMALL PARAMETER IN SECOND-ORDER EL L IPTIC DIFFERENTIAL EQUATIONS
§4.1 Classical case. Problem statement
§4.2 The generalized Levinson conditions
§4.3 Averaging principle
§4.4 Leaving a domain at the expense of large deviations
§4.5 Large deviations. Continuation
§4.6 Small parameter in problems with mixed boundary conditions
V. QUASI-LINEAR PARABOLIC EQUATIONS WITH NONNEGATIVE CHARACTERISTIC FORM
§5.1 Generalized solution of Cauchy’s problem. Local solvability
§5.2 Solvability in the large at the expense of absorption. The existence conditions for derivatives
§5.3 On equations with subordinate non-linear terms
§5.4 On a class of systems of differential equations
§5.5 Parabolic equations and branching diffusion processes
VI. QUASI-LINEAR PARABOLIC EQUATIONS WITH SMALL PARAMETER. WAVE FRONT PROPAGATION
§6.1 Statement of problem
§6.2 Generalized KPP equation
§6.3 Some remarks and refinements
§6.4 Other forms of non-linear terms
§6.5 Other kinds of random movements
§6.6 Wave front propagation due to non-linear boundary effects
§6.7 On wave front propagation in a diffusion-reaction system
VII. WAVE FRONT PROPAGATION IN PERIODIC AND RANDOM MEDIA
§7.1 Introduction
§7.2 Calculation of the action functional
§7.3 Asymptotic velocity of wave front propagation in periodic medium
§7.4 Kolmogorov-Petrovskii-Piskunov equation with random multiplication coefficient
§7.5 The definition and basic properties of the function μ(z)
§7.6 Asymptotic wave front propagation velocity in random media
§7.7 The function μ(z) and the one-dimensional Schrödinger equation with random potential
LIST OF NOTATIONS
REFERENCES

Citation preview

Annals o f Mathematics Studies Number 109

FUNCTIONAL INTEGRATION AND PARTIAL DIFFERENTIAL EQUATIONS BY

MARK FREIDLIN

P R IN C E T O N U N I V E R S I T Y PR E SS

P R IN C E T O N , N E W J E R S E Y 1985

C o p y rig h t ©

1985 b y Prin ceto n U n iv e rs ity Press A L L RIGHTS RE SERVED

T h e A n n a ls o f M ath em a tics Studies are ed ited b y W illia m B ro w d e r, R o b ert P. L a n g la n d s, John M iln o r , and E lia s M . Stein C o rresp o n d in g editors: S tefan H ild eb ra n d t, H . B la in e L a w s o n , L o u is N ire n b e rg , and D a v id V o g a n

C loth b ou n d ed ition s o f Prin ceto n U n iv e rs ity Press book s are printed on a cid -free paper, and bin d in g m aterials are ch osen fo r strength and du rability. Paperbacks, w h ile satis­ fa c to ry fo r personal co lle c tio n s , are not usually suitable fo r lib rary reb in d in g

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(c lo t h )

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Printed in the U n ited States o f A m e r ic a b y Prin ceto n U n iv e rs ity Press, 41 W illia m Street Prin ceto n , N e w Jersey

CONTENTS

PREFACE

viii

IN T R O D U C T IO N I.

3

STO C H ASTIC D IF F E R E N T IA L E Q U A T IO N S AN D R E L A T E D T O P IC S

16

§1.1

16

§1.2 §1.3

P re lim in a rie s

19

T h e W iener m easure S to c h a s tic d iffe re n tia l eq u a tion s

43

§1.4

M arkov p ro ce s s e s and sem i-grou ps of operators

56

§1.5

M easures in the sp a ce o f con tin u ou s fu n ctio n s corresp on d in g to d iffu s io n p ro ce s s e s

73

§1.6

D iffu s io n p ro ce s s e s w ith r e fle c tio n

83

§1.7

L im it theorem s. A c tio n fu n c tio n a l

96

II. R E P R E S E N T A T IO N OF SO LU T IO N S OF D IF F E R E N T IA L E Q U A T IO N S AS F U N C T IO N A L IN T E G R A L S A N D TH E S T A T E M E N T O F B O U N D A R Y V A L U E PR O BLEM S

117

§2.1

T h e Feynm an-Kac formula fo r the s o lu tio n of Ca u ch y ’s prob lem

§2.2

P r o b a b ilis tic re p re s e n ta tio n of the s o lu tio n of D ir ic h le t ’s problem

126

§2.3

On the c o rre c t sta tem en t of D ir ic h le t ’s problem

137

§2.4

D ir ic h le t ’s problem in unbounded domain

147

§2.5

P r o b a b ilis tic re p re s e n ta tio n of s o lu tio n s of boundary problem s w ith r e fle c tio n c o n d itio n s

166

III.B O U N D A R Y V A L U E PR O BLEM S FO R E Q U A T IO N S WITH N O N -N E G A T IV E C H A R A C T E R IS T IC FORM

117

184

§3.1

On p e c u lia ritie s in the sta tem en t of boundary value problem s fo r d egenerate equ a tion s

184

§3.2

On fa c to riz a tio n o f n o n -n e g a tiv e d e fin ite m a trices

188

§3.3

Th e e x it of p ro ce s s from domain

194

§3.4

C la s s ific a tio n o f boundary p oin ts

2 04

§3.5

F irs t boundary va lu e problem . E x is te n c e and uniqueness theorem s for g e n e ra liz e d s o lu tio n s

219

§3.6

T h e H old er c o n tin u ity of g e n e ra liz e d s o lu tio n s . E x is te n c e c o n d itio n s for d e riv a tiv e s

230

§3.7

Secon d boundary va lu e problem

253

v

vi

IV.

CO NTEN TS

SM A LL P A R A M E T E R IN SE C O N D -O R D E R E L L IP T IC D IF F E R E N T IA L E Q U A T IO N S

264

§4.1

C la s s ic a l ca s e . P ro b le m statem en t

264

§4.2

The g e n e ra liz e d L e v in s o n co n d itio n s

278

§4.3

A v e ra g in g p rin c ip le

293

§4.4

L e a v in g a domain a t the expen se o f large d e v ia tio n s

309

§4.5

L a rg e d e v ia tio n s . C o n tin u a tio n

332

§4.6

V.

VI.

S m a ll param eter in problem s w ith m ixed boundary c o n d itio n s

342

Q U A S I-L IN E A R P A R A B O L IC E Q U A T IO N S WITH N O N ­ N E G A T IV E C H A R A C T E R IS T IC FORM

352

§5.1

G e n e ra liz e d s o lu tio n o f C a u ch y ’s problem . L o c a l s o lv a b ility

352

§5.2

S o lv a b ility in the large a t the expen se of a b sorp tion . The e x is te n c e c o n d itio n s fo r d e riv a tiv e s

356

§5.3

On equ a tion s w ith subordinate n on -lin ea r terms

366

§5.4

On a c la s s of system s of d iffe re n tia l equa tion s

381

§5.5

P a ra b o lic equa tion s and branching d iffu s io n p ro ce s s e s

390

Q U A S I-L IN E A R P A R A B O L IC EQ U A TIO N S WITH S M A LL P A R A M E T E R . W AVE F R O N T P R O P A G A T IO N

395

§6.1

Statem ent of problem

395

§6.2

G e n e ra liz e d K P P eq u a tion

403

§6.3

Some remarks and refin em en ts

429

§6.4

Other form s o f n on -lin ea r terms

438

§6.5

Other kinds of random m ovem ents

447

§6.6

Wave fron t propagation due to n o n -lin e a r boundary e ffe c ts

459

§6.7

On wave fro n t propagation in a d iffu s io n -re a c tio n system

466

VII. W AVE F R O N T P R O P A G A T IO N IN PER IO D IC A N D RANDOM MEDIA

478

§7.1

In tro d u ctio n

478

§7.2

C a lc u la tio n of the a c tio n fu n c tio n a l

481

§7.3

A s y m p to tic v e lo c ity of wave fron t propagation in p e rio d ic medium

488

§7.4

K o lm o g o ro v -P e tro v s k ii-P is k u n o v equ a tion w ith random m u ltip lic a tio n c o e ffic ie n t

498

§7.5

Th e d e fin itio n and b a s ic p rop erties of the fu n c tio n p (z )

500

§7.6

A s y m p to tic wave fron t propagation v e lo c ity in random media

514

§7.7

Th e fu n ctio n p (z ) and the on e-d im en sion a l S ch rod in ger equ a tion w ith random p o te n tia l

525

C O NTE N TS

LIS T OF N O T A T IO N S REFERENCES

PREFACE

With every second-order elliptic differential operator

L , one can

associate a family of probability measures in the space of continuous functions on the half-line.

This family of measures forms the Markov

process corresponding to the operator of the operator

L . If one knows some properties

L , it is possible to draw conclusions about properties of

the Markov process.

And conversely, studying the Markov process one

can obtain new information concerning the differential operator. This book considers problems arising in the theory of differential equations.

Markov processes (or the corresponding fam ilies of measures

in the space of continuous functions) are here only a tool for examining differential equations.

As a rule, the necessary results from the theory of

Markov processes are given without proof in this book.

We restrict our­

selves to commentaries clarifying the meaning of these results.

There are

already excellent books where these results are set forth in detail, and we give references to these works. The probabilistic approach makes many problems in the theory of differ­ ential equations very transparent; it enables one to carry out exact proofs and discover new effects.

It is the latter—the possibility of seeing new

effects—which seems to us the most significant merit of the probabilistic approach. This book is intended not only for mathematicians specializin g in the theory of differential equations or in probability theory but a lso for sp ecialists in asymptotic methods and functional analysis.

The book may

also be of interest to physicists using functional integration in their research.

ix

PR EFACE

The two years I have spent writing this book were very hard, I would even say desperate, for me and my family. thank my colleagues for their support.

And I am glad to be able to

I have been happy to see convincing

evidence of the high moral standards of many colleagues.

I especially

wish to express my gratitude to E. B. Dynkin for his constant attention and concern about a ll our problems. F in ally, I must say that this book would never be brought into the world without the enormous labor of my wife, V aleria Freidlin, in her editing, translating and retyping the manuscript.

I feel even awkward

about thanking her for this labor; in essence, she w as my co-author.

M A R K F R E ID L IN

Functional Integration and Partial Differential Equations

IN T R O D U C T IO N

It was known long ago that there is a close relation between the theory of second-order differential equations and Markov processes with continu­ ous trajectories.

A s far back as 1931, the parabolic equations for transi­

tion probabilities were written down in the article of Kolmogorov [1]. Still earlier, these equations on the theory of Brownian motion appeared in physics literature (E instein [1 ]).

It was also established that the mean

values of some functionals of the trajectories of diffusion processes (as functions of an initial point) are the solutions of boundary value problems for the corresponding elliptic differential equations. For a long time the connection between Markov processes and differen­ tial equations was used mainly in one direction:

from the properties of the

solutions of differential equations, some or other conclusions on Markov processes were made.

Meanwhile, probabilistic arguments in problems of

the theory of differential equations played at best the role of leading reasoning.

T h is may be explained by lack of direct probabilistic methods

for studying diffusion processes.

Even the construction of such a process

with given characteristics was carried out with the help of the existence theorems for the corresponding parabolic equations. For the last quarter of a century the situation has changed in an essential way.

The rapid development of direct probabilistic methods for

examining Markov processes allowed one to construct and study them with­ out turning to partial differential equations.

Conversely, the construction

and analysis of the trajectories of the corresponding diffusion process via direct probabilistic methods, enabled the solutions of differential equations to be constructed and the properties of these solutions to be examined.

3

4

IN TR O D UC TIO N

It is not for the first time that such a situation arises in the theory of differential equations.

For example, recall the mutual relations between

differential equations and the calculus of variations.

Originally, the differ­

ential equations served as the means of seeking solutions of extremal problems.

With the development of the direct methods in the calculus of

variations, the possibility appeared of constructing and studying the solutions of differential equations as the extremals of the corresponding functionals.

Similar mutual relations have now been established between

the theory of differential equations and that of diffusion processes. Speaking somewhat inaccurately, one can say that, in the theory of second-order parabolic and elliptic differential equations, the trajectories of diffusion processes play the same part as characteristics do for firstorder equations.

Just as the theory of characteristics makes first-order

equations geometrically descriptive, the probabilistic considerations make transparent many problems arising in the theory of second-order elliptic and parabolic equations. Sometimes the probabilistic methods play the role of a tool for deriving delicate analytical results. of some analytical theory.

Sometimes they are a basis for the extension However, in my view , the greatest value of

such an approach consists in its visualization which turns this approach into an especially helpful instrument for discovering new effects, for a deeper qualitative understanding of the c la s s ic a l objects of mathematical ana lysis. Among the tools of the direct probabilistic research of diffusion pro­ c e sse s, one should, first of all, mention stochastic differential equations. The theory of such equations, originating in the works of Berstein, was basically founded by Ito and (independently) by Gihman, and then has been developed by a number of mathematicians.

The stochastic integral

introduced by Ito, Ito’s formula, and the generalizations of these notions play the central part in the whole theory.

The present state of the theory

of stochastic differential equations is described in the monograph of Ikeda and Watanabe [2]; references to the original works can be found there too.

5

IN TR O D UC TIO N

A s another important factor permitting the direct study of diffusion processes, one should mention the convenient general concept of Markov process and Markov family introduced by Dynkin [1], [3] as w ell as the detailed analysis of the strong Markov property.

The wide use of the

theory of one-parameter semi-groups due to F eller is also worthwhile noting. The theory of martingales serves as a highly suitable instrument for examining Markov processes (see Doob [1], Delacherie and Meyer [1 ]). The transformations of Markov processes, in particular, those involving an absolutely continuous change of measure in the space of trajectories, are also very useful tools which enable one, in a transparent and explicit fashion, to understand the effects of potential terms and first order terms. This leads to an understanding of the affects of these terms on the behavior of the solution of the differential equation. The last ten to fifteen years have seen a development of limit theorems for random processes—central limit theorem type results as w e ll as theorems on the asymptotics of probabilities of large deviations.

In

particular, the counterpart of the asymptotic L aplace method for functional integrals pertains to the results of that kind.

These results proved to be

highly useful in a great number of problems in differential equations which have waited long to be solved. The application of the probabilistic methods for examining differential equations is usually based on the representation of the solution of these equations as the mean value of some functional of the trajectories of a proper diffusion process.

The mean value of a functional of the trajectories

of a random process may be written down as the integral of the correspond­ ing functional on the space of functions with respect to the measure in this space induced by the random process.

This is why such representa­

tions of solutions are sometimes called the representations in the form of functional integrals. The construction of the diffusion process corresponding to the differential operator

6

IN TR O D UC TIO N

(1 )

with the non-negative definite matrix

(a ^ (x )) , is carried out with the

help of stochastic differential equations.

The Wiener process

Wt , the

simplest of the non-trivia 1 Markov processes serves as a starting point. By a Wiener process (one-dimensional), we mean a random process Wt = Wt(u)), t > 0, having independent increments and continuous trajec­ tories (with probability

1 ), and for which

EW^. = 0,

EWj? = t ( E

being

the mathematical expectation sign). It is established that such a process does exist and its finite­ dimensional distributions are Gaussian. the random variable

In particular, for every

Wt(cu) has the density function

t > 0,

(2 n t)~ 1//2exp j - j j : j >

- oo < x < °o. This process is connected, in the closest way, with the operator

1 d2 —— and with the simplest heat conduction equation. 1 dx 2

For

instance, the solution of the Cauchy problem

x ) = g ( x ) ( ot

Z

(2 )

dx

may be represented in the form

This assertion is checked by direct substitution into equation (2). Just as any random process, the Wiener process induces a measure in the space of functions.

In the present case, it is a measure in the space of

continuous functions on the half-line

t > 0 with the values in R 1 . This

measure is referred to as the Wiener measure.

It plays the principal role

in a ll the questions to be considered in this book.

The first construction

of this measure was published by Wiener in 1923 [1].

Later on the Wiener

process and the Wiener measure have been studied in detail.

7

IN TR O D UC TIO N

An ordered collection of r independent Wiener processes (W^,

W£) = Wj. is termed an r-dimensional Wiener process.

Such a

process is connected with the L ap lace operator in R r . What process corresponds to the operator

L

in (1)?

Let us assume for a moment that

the coefficients of the operator are constant: Denote by

a = (o \ )

a matrix such that

a ^ (x ) = a 1^ , b 1(x ) = b 1.

era* = (a 1^) and consider the

family of random processes

X * = aW|. + bt + x, x e R v, b = ( b 1, *••, b r), t >

0 .

(3)

It is not difficult to find the distribution function of the Gaussian process X * and

then to check that

u(t,x) = E g (X *)

is the solution of the Cauchy

problem | = L u (t ,x ),

u(0,x) = g (x ) ,

(4 )

for any continuous bounded function g ( x ) . Therefore, the random process (3) is associated with the operator

L

with constant coefficients.

It is natural to expect that, in the vicinity of every point x e R r , the process corresponding to the operator L

with variable (sufficiently

smooth) coefficients, must behave just as the process corresponding to the operator with the constant coefficients frozen at this point x . On the basis of this reasoning, for the family of the processes to the operator L

X*

corresponding

with variable coefficients, we obtain the differential

equation d X tx = a ( X * ) d W t + b (X x)d t, X x = x ,

where the matrix

cr(x) is such that

(5)

o ( x ) a * ( x ) = (a ^ (x )) ,

b(x) = ( b 1 (x ), ■■•, b r(x )) . If the trajectories of the Wiener process were differentiable functions or at least had bounded variation, then equation (5 ) could be treated within the framework of the usual theory of ordinary differential equations.

But,

with probability 1, the trajectories of the Wiener process have infinite variation on every non-zero time interval. Therefore, equation (5 ) should

8

IN TR O D UC TIO N

be given a meaning.

Ito’s construction is most convenient for this.

This

construction is given in the beginning of Chapter I. One can demonstrate that, under mild assumptions on the coefficients, equation (5) has a unique solution

X * . The random functions

X * , x eR v ,

together with the corresponding probability measure, form a Markov family connected with the operator L .

A solution of C auchy’s problem (4 ) may

be written in the form u(t,x) = E g (X ^ ). A solution of D irich let’s problem for the operator

L

may a lso be

represented in the form of the mathematical expectation of some functional of the process

X * . For example, if D

a smooth boundary

dD

is a bounded domain in R r with

and the operator L

does not degenerate for

x e D U dD , then the solution of the Dirichlet problem

(6 )

L u (x ) = 0,x £ D ; u(x)|^D = ^ (x ) ,

where ifs(x)

is a continuous function on dD , may be written as follow s

u(x) = E«A(X^x) .

(7 )

Here rx = inf { t : X * / D| is the first exit time of the process

X*

from

the domain D . If the term with a potential v

is added to the operator

solutions of various problems for the operator

L + v

sented in terms of the trajectories of the process

X.

L , then the

may also be repre­ For example, the

solution of the Cauchy problem

duCEx) -- L u(t,x) + v (x )u (t,x ), u(0,x) = g(x )

(8 )

is given by the Feynman-Kac formula

Notice that equation (5 ) may be looked upon as the mapping of the space

C Q^

(R r) of continuous functions on the half-line with values in

IN TR O D UC TIO N

R r , into itself:

9

I : W. -» X * . This mapping is defined a.e. with respect

to the Wiener measure in C „

0, OO(R r). 7 The value of X?t at time t is defined as a functional of the Wiener trajectory in the interval [ 0,t]

which depends on x as a parameter:

X * = Ix (Wg , 0 < s < t ) ( t ) . This

mapping allow s formulae (7 ) and (9 ) to be rewritten in the form of integrals with respect to the Wiener measure. Chapter I describes the construction and properties of the Wiener process.

The necessary information on stochastic integrals, stochastic

differential equations, and Markov processes and their transformations is given here. w ell.

Some limit theorems for random processes are included as

In particular, we provide the definition and properties of the action

functional related to the Lap lace type asymptotics for functional integrals. In short, Chapter I introduces those notions and methods which are necessary for the direct probabilistic ana lysis of processes (measures in the space of functions) connected with differential operators. Today there are a number of monographs presenting these results in detail.

A lso , in this book, random processes are a tool rather than an

object of research themselves.

For this reason the results of Chapter I

are, as a rule, cited without proof.

We restrict ourselves to short com­

ments and references. In Chapter II, the formulas representing the solutions of differential equations in the form of functional integrals (in the form of the mean values of the functionals of the trajectories of the corresponding pro­ c e s s e s ) are studied.

B esid es formulas (7 ) and (9), this chapter gives

representations for the solutions of the second boundary value problem as w ell as some other problems. t

The behavior of random processes as

po is a traditional subject of probability theory.

related to problems concerning the stabilization, as

This is closely t -» oo, of the solu­

tions of C auchy’s problem as w e ll as of mixed problems.

It is also

related to the statement of boundary valued problems in unbounded domains. These questions are also considered in Cha pter II. Speaking somewhat inaccurately, one can say that a solution of the first boundary

IN TR O D UC TIO N

10

value problem is unique if and only if the trajectories of the corresponding diffusion process leave the domain D

with probability 1.

Hence the

question of the correct statement of the problem in an unbounded domain is closely related to the behavior of the trajectories as

t -> oo.

If, with

positive probability, the trajectories go to infinity without hitting the boundary of the domain, then supplementary conditions at infinity are required to single out the unique solution.

For example, the Wiener

process in R 2 does not run to infinity, and so the solution of the exterior Dirichlet problem for the Laplace operator in R 2 is unique in the class of bounded functions.

Meanwhile, the Weiner process in R r , for r > 3 ,

goes to infinity with positive probability, and hence, when considering the exterior Dirichlet problem for the Laplace equation in these spaces, one must in addition define the value of the limit of the solution at infinity. In the case of equations of a more general form, “ the boundary at infinity” may have a more complicated structure. Everything depends on the final (i.e . as

t -*

do )

behavior of the trajectories of the corresponding diffusion

process. Probabilistic methods have proved to be greatly effective in examining degenerate elliptic and parabolic equations. these questions.

Chapter III is devoted to

If the coefficients are Lipschitz continuous, then

existence and uniqueness theorems are valid for equation (5 ) regardless of any degeneration of the diffusion matrix

(a 1 ^(x)) . This enables one to

examine the peculiarities of the statement of boundary value problems for degenerate equations.

In particular, the behavior of the corresponding

process near the boundary points is in exact agreement with where and how the boundary conditions w ill be taken. After the process corre­ sponding to the operator has been constructed, it is not difficult to prove the existence theorem and to clarify uniqueness conditions.

The

generalized solution is described in the form of functional integral (7). T h is allow s one to examine its local properties.

Under broad assumptions,

the generalized solution turns out to be Holder continuous.

In order to

ensure Lipschitz continuity or smoothness, one should make some special

IN TR O D UC TIO N

11

assumptions. Chapter III clarifies the conditions under which the general­ ized solution is smooth and gives an example illustrating the importance of these conditions.

Roughly speaking, the smoothness of the generalized

solution is due to the relation between the rate of scattering of the trajec­ tories of system (5) starting from close points and the first eigenvalue (generalized) of the boundary value problem.

The rate of scattering of

the trajectories is defined by a number which is expressed in terms of the Lipschitz constant of the coefficients of equation (5). The results of Chapter III, besides being interesting on their own, serve as a b asis for Chapter IV where elliptic equations with sm all parameter in higher derivatives are dealt with.

The analysis of how the solutions of

boundary value problems depend on these parameters reduces to the follo w ­ ing two questions:

first, to analyzing the dependence of the trajectories

of “ ordinary” equation (5 ) on these parameters, and then to examining the dependence of the functional integral on the parameters contained in the integrand.

Here the dependence on the parameters may be understood in a

rather broad sense.

This may be the dependence on the initial point—in

this way Chapter III studies the modulus of continuity and the smoothness of the generalized solutions.

This may a lso be the dependence on various

parameters involved in the operator L + v ( x ) . Here, for example, belongs the problem on the behavior of the solutions of the equations with fast oscillating coefficients and various versions of the averaging principle. The fact that equations (5 ) are not sensitive to degenerations makes the probabilistic approach especially suitable in problems with small parameter in higher derivatives.

Consider the Dirichlet problem in a bounded domain D:

L 6ue(x) = (L jj+ e L ^ u ^ x ) 4

J i

+ i2

2 i,j= l

A ij( x)

-^ 1