Infinite-dimensional elliptic operators and parabolic equations connected with them

Infinite-dimensional elliptic operators and parabolic equations connected with them

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INFINITE-DIMENSIONAL ELLIPTICOPERATORS AND PARABOLIC EQUATIONS

CONNECTED WITH THEM

Yu.L. DALETSKII

Contents

Introduction 1Chapter I. Ell iptic differential operators in infinite-dimensional

space 4§1. Scales of Hubert spaces 4§2. Functional derivatives and differential expressions related

to them 11Chapter II. Measures in scales of Hubert spaces 18

§1. Gaussian measures 18§2. Random variables. Stochastic equations 22

Chapter III. Parabolic equations of the second order 41§1. Equations with constant coefficients 41§2. Equations with variable coefficients 43§3. Self-adjoint elliptic operators 49

References 52

Introduction

By now many results on differentiation and integration in infinite-dimensional spaces [l] - [б] have been accumulated. The next natural stepwould be the systematic discussion of differential equations, but thisdomain has so far received little attention.

In the book of P. Levy [7] differential equations of the first orderare treated rather extensively (true, this is not done completelyrigorously; see also [δ]). In a number of other articles [9] - [ll] indi-vidual differential equations with variational derivatives are investigated;these are for the most part analogues to ordinary differential equations,and in solving them methods are used that are connected basically with thereconstruction of a functional from its derivative.

The discussion of differential equations of the second order, analogousto the classical equations of mathematical physics, appears considerablymore difficult, because it is essentially connected with integration ininfinite-dimensional spaces. Levy [7] introduced a differential operator ofsecond order with variational derivatives, which he called the Laplace

2 Yu.L. Daletskii

operator. This operator, which was further studied by Polishchuk [12] - [14]and Feller [ΐδ], [ΐβ], has a number of parodoxical properties that makeit resemble operators of the first order. In particular, the quantity ofharmonic functionals related to it is excessively large. Lately Shilovhas shown that this operator is actually, in a certain sense, a first-order operator. Some other second-order equations were treated byNovikov [20].

Baklan [17] and Chantladze [18], [19] obtained the solutions of theCauchy problem for a certain class of infinite-dimensional diffusion equa-tions, by using the method of stochastic integral equations [2l]. Ananalysis of these papers has shown that stochastic equations can also beused for a more general class of diffusion equations [22], [23].

In studying differential equations of the second order we could startoff with an analysis of physical phenomena that lead up to them. In anumber of articles of a physical or semiphysical character individual equa-tions with variational derivatives are deduced or merely written down, andsometimes heuristic arguments connected with their solution are given.

However, there is another way, based on finite-dimensional analogies.This way, which is simpler and more acceptable from a purely mathematicalpoint of view, may appear dangerous to those concerned only with theapplications of mathematical results. It is this way that is followed inthis paper. The author is conscious of its limitations and excuses himselfby the argument that in a little known subject apparently we have to lookfor a path in all directions.

In matrix notation we can write a differential expression of the secondorder for a function of η variables in the form

(0.1)J ti

3,h=l

where B= I! bjfe|| is the matrix of coefficients F" — is the

matrix consisting of the second derivatives of F(xx, ..., x

n). The latter

expression makes sense in case F(x) is a functional in an infinite-dimen-sional space and the coefficient В is an operator.

It turns out to be appropriate to consider functionals in a Hilbertscale of spaces £ a (—00 < a < 00). in terms of the scale the class offunctionals F(x) for which the expression (0.1) is finite is simply des-cribed in its dependence on B.

In particular, we can consider examples when !Qa are spaces of sequencesor functions. Then (0.1) becomes either an expression of the form

d2F

or a differential expression with functional derivatives, which may, forexample, have the form

Infinite-dimensional elliptic operators 3

1 1

0 0

where Lt, L

T are differential operators acting on functions of the variables

t and τ, respectively.The first chapter of this article is devoted to an accurate description

of operators of the type (0.1) and to the investigation of some of theirproperties. The same chapter also gives the necessary information aboutnuclear operators in scales of spaces and functional derivatives; extensiveaccounts can be found in [l] - [з], [24] - [2б].

The basic aim of this article is to solve the Cauchy problem for

parabolic equations of the form — = l?(F) with an operator of second order

such as that described above, where the operator coefficient В is positivedefinite.

The simplest case, in which В is constant, is treated in §1 ofChapter III. The solution of the Cauchy problem for this case can be ex-pressed by an integral with respect to a Gaussian measure, concentrated ina Hubert space. The necessary information about Gaussian measures is givenin §1 of Chapter II (for more details see [з]). The reader who is familiarwith this can proceed immediately to §1 of Chapter III.

The rest of Chapter II deals with the construction and study of morecomplicated measures, which are used in the discussion of equations withvariable operators B(x, t). This construction is carried out by means ofthe solution of a stochastic equation of the form

/ t

I (t) = Ho + [ α (ξ, τ) άτ + f Α (ξ, τ) dw (τ), (0.2)Ό ο

in the spaces of the scale ξ»α. This equation is not essentially more

general than that considered in [l7], [18], but the use of a system ofspaces instead of one space permits us to obtain more general results.

A good account of the theory of equations of the form (0.2) in thefinite-dimensional case is contained in the book of Gikhman andSkorokhod [δ]. The transition to the infinite-dimensional case does notlead to any complications in principle. For this reason, in some placeswhere the exposition is completely parallel to that for the finite-dimen-sional case it is given only briefly.

The last two sections of Chapter III deal with the Cauchy problem for

the equation — = Z2(F) with variable operator B(x, t). Under some assump-

tions on the smoothness of the coefficients and their boundedness it is

proved that a solution of this equation exists and is unique. Moreover, the

solution can be constructed from finite-dimensional approximations. Finally

some properties of the operators of type 12(F) are studied.

4 Yu.L. Daletskii

Chapter I

ELLIPTIC DIFFERENTIAL OPERATORS IN INFINITE-DIMENSIONAL SPACE

§1. Scales of Hilbert Spaces

Some well-known facts are briefly set out in this section, in a formsuitable for what follows.

I. Hilbert-Schmidt operators and nuclear operators. Let be aHilbert space and <S

0 the ring of bounded linear operators in it.

An operator 4 ζ @0 is called a Hilbert-Schmidt operator if there is an

orthonormal basis Φι, Φ2 in £) with the property that

This condition is then fulfilled for any orthonormal basis in $ and

O2(A) does not depend on the choice of basis. The set of Hilbert-Schmidt

operators in $Q is denoted by ©2·

The operators of the class ©2 have a number of important properties:

1) ©2 is a complete normed linear space with the norm O2(A).

2) If A is in ©2, so is its conjugate A*, and σ2(Α) = O

2(A*).

3) ©2 is a two-sided ideal in @

0, and if Л £ @

2, Βζ&

0, then

σ2(ΑΒ) < ||β||σ

2(Λ), σ

2(β4) < ||Б||О

2(Л).

4) For any Α £ ®2

w e have the estimate σ

2(Λ) >

5) The operators of the class ©2 are totally continuous.

6) If Л is a symmetric operator of the class (22, then it has a complete

oo

system of eigenvectors Ф^(& = 1,2,...), Лф£= μ^Φ^) with o2(A)= У\ μ%.

k=--l

7) A has a representation Л = U\A\, where U is an isometric operator

and | Л | = (Л*л4)" a symmetric operator, which is in © 2 , whenever Л£©1·LEMMA 1.1. Let Л^(^ = 1,2,...) be α sequence of operators in the

class @0 that converges strongly to the operator A. If /?£@2, then

lim a2{(Ak —A) B) = 0.ft-+oo

J/ t/ie sequence A\(k = 1,2,...) converges strongly, then

limft->-oo

PROOF. We consider the basis 9l t Ф2,... consisting of the eigenvectors

of the operator |β|: |β|φ£ = ЩФк. Then

=: fj μ| J| ( Лт- Л ) ί/φ

Α ||

2. (1.1)

fel

Prom the strong convergence it follows that \\(Am -4)ί/φ^|| < с, and this

00

shows that the series (1.1) is majorized by the convergent series 2 M-ft-


Since each term of (1.1) tends to zero as m -+ oo, its sura also tends to 0.

The second assertion of the lemma follows from the first after going over

to the adjoint operator.

An operator Α ζ @0 is called nuclear if the series consisting of the

eigenvalues of \A\ is convergent: σ{ (A) — 2 ΗΆ·

W e write 3

t for the set

h=l

of nuclear operators in $Q .

We list some properties of the class ©j.

1) 3i is a linear normed space with the norm o1.

2) 3t a 3

2 and а

2(Л)<: Qj^) for iîgj.

oo

3) If for any orthonormal basis cplf φ 2 , . . . in Jg 2 | | ^ 4 φ Α | | < ; ο ο ,fe=l

then Αζ^.

4) If 4£3i and φ^ (fe = 1,2,...) is an arbitrary orthonormal basis in

fi, then the series^ ' oo

S p ^ = Σ (·4ψ*, ΦΑ)A=l

converges absolutely. Its sum (the trace of A) does not depend on the

choice of basis. We have |sp A\ < Sp \A\ = σ^Λ).

5) If Л £ 3 1 ; then Л*^@,.6) If Л £ 3 2 , B£®2, then 4 5 ^ 8 i and Sp AB = Sp ЯА.The inequality OiiAB) < σ

2(Α)σ

2(Β) holds and becomes an equality

when В = A*; oÂA*) = Sp (AA*) = d%(A).7) ®! is a two-sided ideal in 3

0. If А£<&

и Βζ&

0, then

σχ(ΑΒ) 4 \\в\\а

х(А), σ

1(ΒΑ) к \\в\\а

х(А) and Sp (AB) = Sp (BA).

An operator A in £) is called degenerate if its range of values isfinite-dimensional. The set of degenerate operators is dense in the spaces3i and 32.

2. The Hilbert Scale. Let Г be an unbounded self-adjoint positivedefinite operator in satisfying the condition

ΙΙ^ΊΙ<ΐ· (1.2)The domain of definition $

a of T

a(OL > 0) is dense in ip and becomes a com-

plete Hilbert space under the norm ||χ||α = ||Τ

α*|| (χ e ^

a) .

We introduce in £? another norm ||*||_α= ||Τ

αχ||(α> 0) and consider

the space £)_a obtained from Jg by completion with respect to this norm.

The operator T~a is bounded with respect to the norm || · ||._

a and conse-

quently, after closure, can be defined on the whole of ^ ) _a. The domain

of values of T~a so obtained is .<£, and the inverse operator T

a, which

is clearly the closure in ,ξ)-α of Ta, maps ig to $-

α·

In what follows the symbol {21— >S} always denotes the set of linear

bounded operators that map the space 21 to the space 2.

In this notation the following inclusions hold

6 Yu.L. Daletskii

all four operators being isometric.1

The spaces !ga and $Q-

a are conjugate to one another in the sense of

the scalar product in £j. The value of the functional ξ £ £j_a at ψ € -<ί5α

is given by the formula ξ(φ) = (7ιαφ, ?~

αξ). Since Γ"

αξ = T~

al £ §

α for

ξ££, we find that ξ(φ) = (7'αφ, r~

aQ = (φ, Τ

αΤ~

αΙ) = (φ, ξ), and in this

context we use the symbol (φ, <£) instead of £(Φ) (even if Β, £ £>).

Thus, the family of Hubert spaces {^«} (-oo<a<oo) <QO=<Q, SO con-

structed has the following properties:

1) $ac z $ 3 for oo> a > β > -oo, and ||*||

α > \\χ\\β if ζ ζ £

α-

2) -a = €S·Such a family is called a Hilbert scale of spaces. In what follows the

word " Hilbert " is omitted, because other scales do not occur.Τ is called the generating operator of the scale. We introduce two new

spaces:oo oo

."600= Π ·%α, £-00= U $α·α = —oo α=—00

The space £)oo becomes a linear topological space if we introduce in it

the topology of the projective limit of the spaces ,ξ)α. This space is

always countably Hilbertian, because an equivalent topology can be defined

by means of any countable choice of norms ||*||afe, where ot -» 00.

£}-oc can be identified with the space dual to ipoo In what follows,

as a rule we suppose that

^ ^ © 2 · (1.3)

In this case £?«, is a nuclear space. On this account, when (1.3) holds,

the scale {i&a} is called nuclear.

Nuclearity has a number of important consequences, in particular.

1) in the spaces igoo and ^_oo the weak and strong topologies coincide;

2) if Μ is a bounded set in $-«>, then M£<Qa for some a;

3) in joo and UQ-α bounded closed sets are compact.

3. Linear operators in a nuclear scale. Let Л £ {Jgoo—>,<0-oo}. Then there

are indices (Χ, β for which Α ζ {£)a—-> fee}. The norm of this operator is de-

noted by

\\AX\\R

It is not hard to verify that the operator i £ { â- > ^ } can be represented

in the form

(1.4)

where Βζ<50 and

о оWe write A for the restriction of A to ig. A is defined on a set dense

1 In what follows the symbol ~ is frequently omitted where this does not lead to

an inconsistency.


in OQ if A£{<Q<X—> J0e} for β > 0. I f at the same time (X < 0, then byо

(1.4) A i s bounded.о

L E M M A 1.2. Let Α £ {!Q-I—-> i}· Then A is a nuclear operator. If A

is a set of operators for which ||Α||(-ι,ι) < С, then the series

о °°

ft=l

where }φ } is an orthonormal system in $Q , converges absolutely and

uniformly over 21.о

PROOF. The fact that Л is nuclear follows from the representation

Д = Г1 £ Т - 1

, since Т~г^^

2, BT~X£Z2. Also ||#||<C (Л£Я), and so

because of the estimate

, φ Α ) С

the lemma follows.COROLLARY. We consider three operators Alt A?, R with the proper-

ties

О

It is clear that S — AiRA

2 £ {.'Q-i —> 1} and hence S are nuclear operators.

By a projector in the scale {§a} we always mean an operator Ρ that

satisfies the conditions:a) P«6_oo cr £oc;

b) P2 = P.

A projector is called orthogonal if it is orthogonal in § and α-ortho-

gonal if it is orthogonal in jga. The set of all projectors in the scale

{$Qa} is written as SjS, the set of orthogonal projectors as $js

0, and the

set of α-orthogonal projectors as $pa. Clearly, if Ρζ^β, then

T~aPT

a^', also T-

aPT

a£%

a, if Ρ £ %. The symbols «β', denote

the subsets of the corresponding classes consisting of the finite-dimensional

projectors.

The domain of values of a projector Ρ is denoted by 2p. An increasing

sequence of projectors

Λ c= P2 с .. . cz P

n с ...

is called α-complete (or simply complete when а = 0) if it tends strongly

to the unit operator in Jga. It is easily seen that if the sequence

\Pk\ is α-complete, then the sequence {TaP

kT~

a} is complete.

It is not hard to give an example of a sequence of projectors {P } that

is α-complete for all a. Let Ф^ (k = 1,2,...) be the sequence of eigenvectors

η

of T: TqPfe = \±kVk(k = 1 ,2 , . . . ) . Put Pnz= Σ (χ> ΦΑ)ΦΑ· Then our asser t ion

follows from the relation Τ ^ Τ " * = Pn.

Let {Pk\ be an α-complete sequence of orthogonal finite-dimensional

projectors. In each 2P we choose an orthonormal basis. Then we can

8 Yu.L. Daletskii

construct an orthonormal system Φ& (k = 1,2,...) in Jg for which

η

Pnx = 2 (x, ΨΑ) <Pfc· Prom the α-completeness i t follows that the expansion

oo

y= Σ (У, <Ы<р* (^&)ft=l

is strongly convergent in £)a. On applying T

a to this and putting χ = T

ay,

we obtain for any χζ$£ the expansion, strongly convergent in <Q,

oo

*= Σ (*, Γ"αφ*) φ*·

This formula means that { T ^ } forms a basis in £3. It is easy to see that

the sequences {Ταφ&} and {T~

a<Pfe} are biorthogonal. It is known that if one

of two biorthogonal sequences forms a basis, then so does the other

(see [25]). So we obtain the following proposition.

L E M M A 1.3. If a sequence of finite-dimensional orthogonal projec-

tors Pk is Oi-complete, then it is also (-0C)-complete.

We use sequences of finite-dimensional projectors to construct finite-

dimensional approximations to operators. For example, if {Pk\ is a complete

system, then clearly

Ax = \imPkAP

hx (χζ&).

fc-УОО

It is easy to verify that for nuclear operators strong convergence can bereplaced by uniform convergence. We need a more complicated, but more exactassertion.

LEMMA 1.4. Let Λ €{&*-»&}, Л €{£-«-»&*}. 4>€{£-1->Ы.Put A^^PnAiPn, А^^-РпАгРп, R

{n) = P

nRP

n, where the sequence of

projectors PkQ.ô (& = 1, 2, . . .) is complete and CL-complete. Then1

lim aiiAiRA*- A^R™A™) = 0, )

(1.6)П->оо

limП-+ОО

PROOF. I t follows from (1.4) that Ax = TxBxTa, Л2 = TaE2T'x,

R = т~°^Т'а, where B l t B2, Q are operators of the class <S0-Now we estimate each component of the right-hand side of the formula

A,RA2 - 4(

x

n ) Д ( П ) 4 П ) = A[n)RPn (I - Pn) + A[n)R (I - Pn) A2 +

+ РпА, (I-Pn) RA2 + {I-Pn

The f irst of these can be written as a product of several factors

Kn = Pn · T-^B, • TaPnT~a · Q · T~aPnTa · B2 · Г

1 (I - Pn).

1 Here and henceforth we usually omit the null in expressions of the type of 5,

where S = A^M^.


Here Τ~ιΒ^^

2 and by Lemma 1.1 O

2(T~

1(I - P

n)) -» 0. The other factors

are uniformly bounded, and so lim ot(Kn) - 0. In the same way we obtain

n-* oc

the required estimate for the second component on writing it in the form

Pn · Τ~

ιΒ, • T

aPnT~

a ·<?·(/- T~

aPnTa) BzT'

1

and using the relation 02({I - T~

aPnTa)B

2T~

i) -» 0, which follows from

Lemma 1.1. The other components are estimated similarly.

The second relation is deduced in the same way, using only the «-com-

pleteness of the sequence.

REMARK 1. It is easy to see that if the operators Л1 ( A

2, R depend on

any parameters and are uniformly bounded in the corresponding norms, thenconvergence in 1.6 is uniform with respect to these parameters.

REMARK 2. We could consider two scales {£)«} and {$Qa} and suppose

that Λι£{£α->£ι}> Д€{$-а->Ш, Λ€{£-ι->£ι-α}· It is easy to see

that in this case an assertion similar to that of the lemma can be made.We need only postulate that the scale {$

a} is nuclear.

4. Examples. We consider two concrete constructions, which come under

the scheme described above.

a) Let fe = l2 be the space of sequences χ = {x

n\ satisfying the condi-

tion

ιι*ιι»= Σ ыа< « .

Put Tx = {Цп*п1 . where μη is a sequence of non-negative numbers. The con-

dition (1.2) is satisfied if liij > 1 (n = 1,2,...), and (1.3) is equiva-lent to

n=l

The space ,£a consists of the sequences that satisfy the condition

oo

II γ | |2 V | | 2 0 | r I2 <^ OO (\ Ί\

\\x Hot — ΖΔ r n I xn I <- °° · \1·')n=l

It follows that $«> in this case consists of sufficiently rapidlydecreasing sequences that satisfy condition (1.7) for any α however large.Finally, £j_oo consists of the sequences that satisfy this condition forat least one a.

The scale generated by the sequence {μη} will be written &α

μη}• The

operators in such a scale are given by infinite matrices A —1| djk ||}%=ι·

Consider, for example, two scales §« and $i n> and suppose that

Σ=l

10 Yu.L. Daletskii

Then A takes $ ί μ η } into £ i U n } . Foroo

h=i

/ \

ajk

We note that if the matrix ||a/k|| satisfies the condition

Σ (1.9)

for some sequence λ· (j = 1,2,...) such that 2 ~ Г - < ° ° » *Ьеп we can always

choose numbers μ/j for which (1.8) is satisfied. Thus, for matrices thatβη}

obey the condition (1.9) we can always choose a scale &αβη} (not

ίμ η }

) ιλ η }£)ι λ η }}. The adjoint operator thennecessarily nuclear) such that Αζ {$ί

satisfies the condition A*£ {£i\n>—> ^ 1 ι } ·This example is standard. For, let ^ be a nuclear Hilbert scale. Let

\]/fe(fe = 1,2,...) be the set of eigenvectors of the generating operator ofthe scale: Γψ£ = μ ψ£ (k = 1,2,...). This set is an orthogonal basis in any

£ a ; we take i t to be normed in £j: ||%jj = l . Then any vector x£!ga isrepresentable in the form of a series convergent in $ a ,

whereft=l

oo

Thus, the scale in question is isomorphic and isometric with the scale

b) Let G be a bounded domain in the Euclidean space Rn with piecewise

smooth boundary γ. Let D denote the set of 2k times differentiable functions,

satisfying boundary conditions, for instance, y\y= 0, Day|

r= 0 (|a|<fe-l).

Put Ту = (-l)khky + c*y (y e D), where c

2 is so chosen that (1.2) holds.

For Τ we take the closure in —• S2 (G) of this operator. The scale of

ο, ι

spaces §a then consists of the Sobolev spaces Щ

а(С). Provided that k> -n

the condition Г ^ 621 в fulfilled (see [27]).

Similarly we can consider the case G = Rn; however, for the scale to be

nuclear we have to multiply Τ by a sufficiently rapidly increasing function,for example


§2. Functional derivatives and differentialexpressions related to them

I. Functional derivatives. Let 91 and £ be real linear topologicalspaces. We consider a function у = F(x) defined in a neighbourhood U ofthe point χ

οζΈ. and taking its values in £. For any /г£91 and suf-

ficiently small t the expression F(x0 + th) has a meaning. Its derivativewith respect to t, calculated at t = 0, is called the weak differentialof F(x) at x0:

The weak differential is a homogeneous function of h; if it depends onh linearly, then there is an operator F' (x0) £ {91—> £} such thatDF(x0, h) = F'(xo)h. The operator F'(x0) is called the weak derivative ofF(x) at *0.

Regarding F' (x0) as a function with values in {91—>£}, we can define asecond derivative F" (x) £ {91 -> {91 —> £}}.

If the spaces 91 and 2 are normed, we can give another definitionof the derivative. We suppose that there is a representation

F (x0 + h) - F (x0) = Ah f В (h) (xQ + A£E/),

where Л £ {91 -* 2} and !l (/jj ! l — > 0 as \\h\\ -> 0. Then the operator

A = F'(x0) is the strong derivative of F(x). It is not difficult to seethat a strong derivative is also a weak one. It is known that if the weakderivative is uniformly bounded and continuous in some neighbourhood of x,then it is also the strong derivative at this point.

The second strong derivative can be obtained by repeating the proceduredescribed. It can also be constructed in another way. Suppose that we havea representation

F{xo + h)-F(xo)-F'(xo)h = b(xo, hu h2) 4- о (|| h ||2),

where b(x0, h

lt /i

2) is a bilinear function of h

u /^

2£9Ι and takes its

values in S. This function defines an operator F" (x0) £ {91—> {91—>&}},

such that b(xo, hlf h

2) = (F"(xo) /г

1)/г

2.

It is known that F"(x) is continuous, both definitions of the secondstrong derivative give the same result. In all cases below, where this ispossible, the derivatives are understood in the strong sense.

Consider, in particular, the most important case for us: when F(x) isa non-linear functional with a domain of definition in 91. In this case2 = R

l = (— oo, oo) so that /"'(я) £{$—>/?!} = 91·, and finally

Г(*)€{И->И·}.We consider functions у - F(x) defined in igoo or in a domain of this

space. If such a functional is differentiable, then / (z) £ £)_«,, that is tosay, for fixed χ there is an index β(χ) for which F' (x) £ %(*). Below, asa rule we consider cases in which the index β can be chosen independentlyof x.

12 Yu.L. Daletskii

Let Ua a Jg

a. Let C

k(U

a) be the class of functionals that can be con-

tinued to Ua and are strongly continuous in this domain together with their

derivatives of all orders up to k inclusive. Similar classes of vector func-

tions with values in $Qai and operators with values in {§

α ι —> §p} are de-

noted, respectively, by Ca1(U

a) and Οα^βΦα)·

It is natural to ask whether a functional can be reconstructed from

its functional derivative. Let F(x) be a function of the class C1^ ) ,

where U is a simply connected domain in the Banach space Si with values in

the Banach space S. A curve Ζ in И is a vector functionx = x(t) (0 <: i < 1) with values in this space. A curve is said to be smoothif x(t) is strongly continuously differentiable. If Ζ: χ = x(t) is a smoothcurve in U, then the function φ(ί) = F(x(t)) is, obviously, differentiable,

and φ'(ί) = F'(x(t))x' (t). Hence, if *(0) = x0 and *(1) = x

lf then

ι

F (x,) - F (x0) = jj /" (x (t)) x' (t) dt. (1. J 0)

b

The integral on the right-hand side can be interpreted in the sense ofBochner. We call it the integral along Ζ and write

F' (x) dx.ι

The result in (1.10) does not depend on the choice of the smooth curve

in U along which the integration is carried out. In particular, we can take

as part of integration the interval χ = xo(l-t) + x

xt, and then

ι

F(xl)-F(x

0)=^ F' (x

o-\-t(x

i-x

o))(x

1-x

Q)dt. (1.10')

о

Now suppose that the following representation holds:

F(x) = a+\G(x)dx (a = const), (1.11)

where G(x) is a functional in the class C(U) with values in {И — > £ } andthe integral does not depend on the choice of the smooth curve Ζ in the

simply connected domain Ucz'U. Then F(x) belongs to the class C1 (U) and

F'(x) = G(x). For

i

\F(x-\-h)~F(x)~G(x)h\\^\\ \ G{x + th)hdt-G{x)h

th)-G(x)\\dt = \\h\\-o(i).

These arguments lead immediately to the following result.LEMMA 1.5. Let Fn(x) (n = 1,2,. . .) be a sequence of functions with

values in the Banach space β and belonging to the class CX(U), where Uis a simply connected domain in the Banach space ST.


We assume that the following conditions hold:

a) The sequence Fn(x) tends strongly to the function F(x) in U.

b) The sequence Fn(x) tends strongly in U to the function G(x) of the

class C(U) with values in {2(—> 2} and is uniformly bounded in any compact

subset of U.

Then F(x) e C1 (U) and F' (x) = G(x).

To prove this it is sufficient to pass to the limit in the equation

(1.10) for Fn(x) and to obtain a representation of the form (1.11) for

F(x).

REMARK. It is easy to prove similar propositions for higher order

derivatives, by applying the lemma several times.

2. Differential expressions of the first and second order. Let

F(x) efriU), where {/£&». If F(x) is a functional, then F'(x) ££-„>,

and so for every function a(x) with values in £joo defined in U there is a

meaning for the expression l±(F) = (a(x), F'(x)) (x e U), which we shall

call the differential expression of first order of F(x) in U.

If F(x) is a functional of the class Cx(U

a), we may suppose that

a(x) e Ca(U

a).

If F(x) is a function of the class C$(Ua), then F' {x)£{5&

a—>£p} (x£U

A first order differential exprrssion can be defined by the formula

lx(F) = Sp \A(x)F' (x)B(x)} , where the operators A(x) and B(x) are chosen so

that this expression makes sense, for instance,

Now let F(x) be a functional of the class C2(U

a). Its first derivative

is in C2(U

a). The first order differential expression for F' (x) gives rise

to a second order differential expression for F(x):

l2 (F) •= Sp {A (x) F" (χ) Β (χ)} (x£U). (1.12)

Here it is supposed that Α (χ) ζ {<ρ_α—>$J, Β (χ) ζ {§-ι—-> $

α}· In

particular, we can take В = A*. We call the expression so obtained,

l2(F) = Sp{A(x)F"(x)A*(x)}, (1.13)

elliptic.Note that we may consider an expression of a somewhat more general

type than (1.12): Sp {C(x)A(x)F" (x)B(x)}, where C(x)£®0.By analogy with the finite-dimensional case we call the expression

F'(x) = grad F(x) the gradient of the functional F(x), and the expressionSp F' (x) = div F(x) the divergence of the vector function F(x). In thiscontext it is natural to call the expression

Sp/'"(a;)=divgrad/I(a;) (1.14)

the Laplace operator on the functional F(x). The expression (1.14) has ameaning if F(x) e C2(U_1).

3. Finite-dimensional approximation. We call a functional F(x) definedin a domain U С $«> cylindrical if for some projector

F(x) = F(Px). (1.15)

14 Yu.L. Daletskii

A cylindrical functional can be continued automatically to those ж£$-ооfor which Px e U. Below we consider only those cylindrical functionals thatare defined on the whole of ig-oo and satisfy (1.15) for some Ρ£^βό·

>Гпе

set of such functionals is denoted by fto.Let F(x) e K

0 be twice differentiable

F(x + h)-F(x)^(h,F'(x))+^(F"(x)h,h) + o(\\h\\D. (1.16)

The left-hand side of this equation vanishes for Ph = 0. Hence

F' {x) = PF' (x), F" {x)=PF" (x) P. (1.17)

We choose an orthonormal basis ek(k = 1,2, .... n) in Sp. The functional

F(x) reduces to a function of η variables

f(xux

2, ···, x

n)= ^ ( 2

xk

eh) t Xh = {

xi

ek)· It follows easily from (1.16)

that

2 ёfc=l ft=l

Using (1.18) we find the form of the differential expressions

(F) = (a, №') = 2 a* -g-, a, = (α, βΛ).

ft=i A

Consider a functional F(x) of the class £($«)· Let Ρ£$βό· ê functional

Fp(x) = F(P*) is cylindrical. If Pfe (fe = 1,2,...) is an <X-complete sequenceof projectors, then by continuity

F(x) = \imFP (x) (x£Q

a).

h-yoo n

Thus, the continuous functional F(x) can be approximated by functionalsof the class тс

0, that is, actually by functions of a finite (though in-

creasing) number of variables. We also note that this convergence is uni-form on any compact set Μ cz §

a, because the sequence Pfe converges uni-

formly on such a set and F(x) is uniformly continuous.

We now see how differential expressions of the first and second orderbehave in such approximations.

T H E O R E M 1.1. Let F(x)£C*($a) and let the sequence Ρk (fe=l,2,...)

be OL-complete. Further, let

α (χ) ζ Ca (g

a), Α (χ) ζ Cai (&a), В (x) e C-

it _

a (Ы-

Then the following assertions hold:


1) For s - 1 lv (F) = lim lt (FP), h (F) = lim Zf * (FP ), «Леге Zi (F) =fc-юо ft->oo

2) For s-2 L(F)= lim 12(FP), l2 (F) = lim f2

h (FP), where 12(F) =h—юо h-+oo

= Sp A (x) F" (χ) Β (χ), llk (F) --= Sp A (Pkx) F" (χ) Β (Pkx).

If in addition the sequence P^ (k = 1,2,. . .) is complete, then

l2 (F) = lim Sp [PkA (Pkx) PkF"Pk (x) PkB (Pkx) Ph]·

In all these relations the convergence is uniform on any compact set

Μ a $Qa·

PROOF. 1) It is not difficult to show that Fp(x) =PF'(Px). Therefore

1Z, (F) - h (Fph) | = | (a (x), F' (x) - PhF' (Phx)) | <

< II a (x) \\a { || (I-Pk) F' (x) ||_e + || F' (x)-F' (Pkx) | |_ a}.

Here the first component tends to zero because Pk -* I, and the secondbecause of the continuity of F'(x).

p.The relation h(F) = limii (Fp

k) is proved similarly.

2) The relation Fp(x) = PF"(Px)P holds. Hence

l2 (F) - 1

2 (F

P) | = | Sp μ (яг) F" (χ) Β (χ)-Α (χ) PF" (Px) PΒ (χ)} | <

< oj {А (х) F" (χ) Β (χ) —Α (χ) PhF" (Λ) P

hB (χ)} +

+ σ, {Λ (χ) Pk [F" (χ) - F" (Px)] P

hB (χ)}.

Here the first component tends to zero by Lemma 1.4 and the second by

Lemma 1.1 and the continuity of F"(x).

The proofs of the other relations are similar.

Arguments similar to those that precede the theorem prove the uni-

formity of convergence on compact sets.

1. Examples, a) We consider the scale ^α

μ η satisfying the condi-

oo

tion 2 -v-<oo· ТЬб functionals defined on the spaces of this scale are

representable as functions of a countable number of variables

A derived functional of class C1 ($Q

a) is defined by a sequence

dF dF dFdx\ ' dxo ' dx

n '

16

satisfying

Yu.L. Daletskii

OF

dx.<oo.

The second derivative of a function of class C- (£>a) is defined by а

matrix. dx

k

such that the matrix generates an

operator that is bounded in Z2.

A first order differential expression has the form

= (α, F ) = 2 J «ftft=l

dFdxk '

ft, « = i

and

ft, s = l

ft, s = l j = i

where

If A = И ay fell, B= ||byfe||, then the second order differential expres-sion (1.12) takes the form

{ Σ ι ^ft, s = l

A

Let Pn be a projector defined by the formula

Pnx = (x

lt x

2, ..., x

n, 0, 0, ..., 0, . . . ) .

It follows from Theorem 1.1 in this case that

= lim

ft, S=i

Σj==l

b) As another example we consider a scale consisting of spaces

that is generated in a domain G of Rn by the operator

Ty(x) = (-l)feAfey + c

2y with the boundary conditions

The elements of the space of this scale are ordinary or generalizedfunctions. The function (possibly generalized) defining a derivative F' (y)of the functional F(y) is called the variational derivative and is written


. The differential expression Zi(F) in this notation takes the form

h (F (y)) = (a (y), F' (y)) =[a(x, y) -j^— dx.G

The operator / ? £ { $a—

> $p} corresponds to a nucleus

(possibly generalized) such that

(Ry) (χι) = ξ R (χι, x2) У (xz) dx

2.

G

If β > α + 1, this nucleus satisfies the condition

\ \ | R (Xi, x2) |

2 dx

{ dx

2 <C oo.

G G

The nucleus that corresponds to an operator F" (y) is called the

second variational derivative of the functional F(y) and is denoted by the

symbol

In this notation a differential expression l2 takes the form

&2fA ^ ' ^

2 ; "бу (до) by (χ

3) B ^

3'

ι ; y)

dXl d X l dXz' ^

1 Л 9^

G

Here the kernels A(xl f x

2; y) and В(л;

1( х

2; у) may be generalized functions.

Particular cases of (1.19) are the expressions

dx (Laplace operator), (1.20)J by (χ) by (χG

wG

where K(xlt x

2) is a continuous kernel.

Finally we can consider an expression of the formгс б

2 F

\ \ K(xu x2; y) LX1LX2 6y{Xi)6y{Xz) dx, dx2, (1.22)

where L . is a differential operator. If this operator acts from ig-i to

§a, then for the expression (1.22) to make sense it is sufficient that

the functional F(y) is differentiable in iga.

We remark that if F(y) is differentiable in £jb then its second

variational derivative is, in general, a generalized kernel. Suppose, for

example, that

by (χ,) by (x2) ~

T ^ ' ^ --' '

where φ(χι) and ψ(*ι, л:2) are continuous functions. In this case the

expression

\ φ (χ) dx,

G

18 Yu.L. Daletskii

which was introduced by P. Levy as a Laplace operator AF, makes sense.This operator has been discussed in the introduction. It cannot be obtainedfrom the formula (1.12), but as the limit of similar expressions.

Chapter I I

MEASURES IN SCALES OF HILBERT SPACES

§1. Gaussian measures

This section sets out briefly some well-known facts of the theory ofmeasures in linear spaces (for details see [з]).

I. The finite-dimensional case. A set Ω with a distinguished algebraβ of subsets is called a measurable space {Ω, £}.

A measure space {Ω, £,μ} is a measurable space {Ω,£} in which an

additive non-negative function is defined on £,the measure μ. When £

and measure μ are countably additive, {Ω, £, μ} is called a space with a

σ-measure.

In what follows Ω is always supposed to be a linear space.

Consider first the case when Ω is the Euclidean space Rn and £ a

Borel algebra of subsets in it.

Let a e Rn, and let В be an invertible positive definite operator in

Rn. We define a measure by the formula

а), (x-a))}dx (<?££). (2.1)

Such a \i3t a is called a (non-degenerate) Gaussian measure with mean

value α and with correlation operator B. A degenerate Gaussian measure in

Rn is obtained if we consider the Gaussian measure on a subspace L С R

n·

The function

is called the density of the Gaussian measure. The function

χ{χ)=

Rn

is called the characteristic function of the measure μ.

For a Gaussian measure (2.1) the characteristic function has the form

1B, a(z)=-exp | _ _ ( # £ , x)+i (x, аЦ . (2.2)

Note that this expression makes sense even when В is not invertible. Inthat case a degenerate Gaussian measure corresponds to i t .

The following properties can be verified by direct calculation:

a) μΒ,α(Ηη) = ί, ξ χμΒ, a(dx) = a, J (χ, φ) (χ, ψ) μΒ, α (dx) =Rn Rn


b) Let Rn = R

n @ R

n.m and χ = (χ', χ"), where χ

1 e R

m, χ" e R

n-m.

Then the measure in the space Rm defined by the formula

μ(ζ)) = μβ,α(()χ Rn-

m) (Q С R

n) is Gaussian with mean Pa and correlation

operator PBP, where Px = x'.

c) Let f(x) be a continuous function satisfying the condition

fix) 4 Cea(B 1 χ

·χ ) (Ο 0, α > 0). Then the function

g(*,t)= \ f(y)VBt,x{dy)^ ^ f{x-y)VBt,Q{dy) (2.3)

fin Rn

satisfies on some interval 0 < ί < ίο the differential equation

dg 1 - ^ , <32g . „ .,τ — c\ / ι U i h ~~ϊ Μ \ /

Ot Δ •*—I J OX j OXfr

with the condition gix, 0) = fix).

2. Gaussian measure in a nuclear scale. Let Ρ be a finite-dimensional

projector that projects ^_oo into <£χ and is orthogonal to $Эо(Р € -έό)·

A set of the form Qa(P, Μ) = {χζί£

α; ΡχζΜ}, where Μ is a Borel set in

<£P, is called a cylindrical set with generating operator P, base Μ and

support p.

These characteristics of a cylindrical set are not defined uniquely:

they can always be extended. For let Ρ cz Ρ' £ anc

* let

Μ ' = {χζ Xp>; Px ζΜ) be the complete inverse image in XP> of Μ under the

map PXp-~>Xp. Then clearly QaiP', M') = QaiP; M).

The family of cylindrical sets of £ja is denoted by £«. The class

2a is an algebra of sets. For any finite number of cylindrical sets can

be supposed to have a common support, and hence everything reduces to

operations on Borel sets in a finite-dimensional space.

The minimal σ-algebra in $Qa that contains £

a is written as S

a.

Let us suppose that a measure μ is given on the measurable space

{£)<*> Sa} . This automatically defines measures in the spaces XP if we

take the measure of a cylindrical set to be the measure of its base.

The measures μ' ' so obtained must be compatible among themselves:

μ(ρ<)

{Qa {P\ Μ')} = μ(ρ) {Q

a (P, M)}, where M' - {χ ζ %

Ρ.; Ρ

χ ζ Μ}, Ρ cz Ρ'.

(2.5)

Conversely, if in the spaces Xp a set of measures satisfying the con-

dition (2.5) is given, then we can define a measure on Sa by putting

ViQaiP, Μ)) = μ(Ρ)(Λ/).

We consider, in particular, Gaussian measures. Let 5(φ, ψ) be a

positive definite quadratic form in <£)«> and let a££)_oo. In each subspace

XP we consider a Gaussian measure with correlation operator Sp, generated

by the form S (φ, ψ) = (£Ρφ, ψ), (φ, ψζΧρ), and the mean Pa. Since

Ρ CP'Sp = PSpiP, by property b) of the last subsection the condition of

compatibility (2.25) is fulfilled.

оThus, for any de (-00,00) we obtain a measure space ($

a, £«, μβ, a)·

20 Yu.L. Daletskii

The question arises naturally whether this measure can be extended tomake it σ-additive on the σ-algebra 2

a. This question has been solved for

arbitrary measures on cylindrical sets (see [з]). For the case of Gaussianmeasures the solution is exceptionally easy, and we give here a shortproof of the results of interest to us, adapted to our terminology.

First of all we give some preliminary arguments. For the calculationof correlation forms it is convenient to use the easily verifiable relation

(z-Pa, q>)(z-Pa, ψ)μ»4 (<te) = S(q>, ψ). (2.6)

Here the integration is over any finite-dimensional subspace £P = P!Q

that contains the vectors φ and ψ.

Now let μχ,α be a Gaussian measure on {^-1, S-i} with correlation

form θ(φ, ψ) = (φ, ψ) and A £{£)-ι —>$γ}· Then у = Ax is a measurable map

of 2-i on £°. This map induces a measure μ^,Λα in {£jv, S°}, which is a

Gaussian measure (this fact need only to be verified for maps of finite-

dimensional subspaces). Using (2.6) we find the correlation form of the

measure MvS,Aa:

= \ (Αχ-

Яр

= ^ (χ — α, Α*ψ)(χ — α, Α*ψ) μ<ρ>(dx) = (A*q>, 4*ψ). (2.7)

Xp

LEMMA 2.1 . Let MvS,Aa be a Gaussian measure on {j§Y, SY} with meanAa and correlation functional 5(φ, ψ) = (Л*ф, Λ*ψ).

If « € ^ - ι and Α ζ {ig-i—>^γ}? the measure [Ls,Aa n a s a ^-additive con-tinuation on {$Qy, 2y}.

PROOF. We consider the simplest case f i r s t , that in which A = I,Υ = - 1 . As i s known, i t i s suf f ic ient to show that for any ε > 0 there isa bal l Од in Jgv with the property that for any cyl indrical set Q(P, M)outside σ^ we have m / f l (Q) < ε · Let ад = [χ: \\χ - α | | - ι < R\. Since

Vl,a(Q(P, Μ)) = μ|Ρ)

α(Α/), where Ma{x£XP: \\ x — Pa | | _ t > Щ, we have

μ/, α (Q (Ρ, Μ)) = μ(^ (dx) < — 1 ^ . J Γ Ι Π*-

Ρα"2 <

M (2π)

2 Н

, 1< \Τ l(x-Pa)\\2e ^ = ^ . <

(2π) 2 Д ^

For sufficiently large R we obtain the required result.Now we consider a map у = Ax of the space Jg-i to £)v. It is a measur-

able map of {i§-i, й-i} to {£)v, S

Y} and induces a σ-measure in the latter.


This measure has, by (2.7), the correlation functional (Α*φ, Λ*ψ) and mean

Aa. On £Y it coincides with 14s, Да-

REMARK. Inasmuch as under the conditions of the lemma Us,/4a is an

image of \î>a under the map у = Ax, integration with respect to lAs

;4

a

c a n

be reduced to integration with respect to \iji(1', in other words, we have

the formula

\ ί(Λχ)μΐ,

α(άχ), (2.8)

if f(Ax) is integrable. In calculating integrals of the type (2.8) for a

continuous function we can use the following method: f(x) is approximated

by cylindrical functions fp(x) = f(Px). An integral of a cylindrical func-

tion reduces to an integral on subspaces X P, and if any conditions for

passage to the limit under the integral sign hold, then

\ f(x)ii(dx) = hm \ ΙΡ(χ)μ(άχ)^ηπι \ fP(x)[i^(dx). (2.9)

For instance, consider the function

/ (x) ---.= £?alWI2_, = gallT-^H2,

It is continuous in ,^-i and is the limit of the monotonically increasing

sequence of functions

— 2 ~

/ P (дг) — eajiT-v f l.v||a ., e k=iKk ^

where Pn are projectors that commute with the operator T.η

Pnx--= Σ ^ΑΨΑ. ^φΑ=^λΑφΑ.A = l

Using (2.9) we find 1 for t > 0

(2л 0 2 Jfp

n

lim ΓЛ=1

If we now make the change of variable у = Ax, where -I £ {λ> . ι —> £jv}.

then by (2.8) we get the formula

« ΐ Ν Ι ΐ

— d e t ( 7 — α / 7 ' " 2 ) - < οο f o r 1 > < α / < 1 . ( 2 . 1 0 )

1 A similar formula was obtained earl ier by Shirokov.

22 Yu.L. Daletskii

§2. Random variables. Stochastic equations

I. Random variables with values in a scale. A probability space is a

space with σ-measure {Ω, S, P) satisfying the condition Ρ(Ω) = 1. The

points of Ω are called elementary events, the elements of the σ-ring S

events, and the values of the measure on these sets probabilities of the

events.

Let {Ω, Si} be a measurable space. The measurable map £(ω) of Ω to

Ω! is called a random variable with values in Ωι. If a random variable

<ζ(ω) is given, the {Ω1? S

t} automatically becomes a space with σ-measure

μ (Μ) = P {ω: ξ (ω) £ Μ} == Ρ {ξ (ω) £ Μ}.

Thus, any random variable corresponds to a normalized σ-measure on the

set of its values (probability distribution). Conversely, if in {Ω, 8J

a normalized measure μ is given, we can always take it to represent the

probability distribution for a random variable £(ω) = ω, on identifying

п± with the space of elementary events.

A choice of a finite number of random variables «^(ω) £η(

ω) with

values in {Ωΐ5 Si} generates a measure in the product space

μ (Μ) = Ρ {ξι (ω), . . ., ξη £ Μ

η), where M

n ζ S

t X g

t χ . . . χ S

t.

η

A set of random variables £t(0< t< T

o) with values in a space Ω

α is

called a stochastic process in this space. (The probability space

(Ω, S, P) is supposed to be the same for all £). The set of measures

(О </!</,< ...<*„< Го, w = l, 2, ...)

is called the set of finite-dimensional distributions of the stochastic

process in question. This set is compatible in the sense that

Ρ*1· • -'A-IVA+I- · Ап (

Л* > ; · · *

X '

4ft-l

X A* X Â+l

X · · '

X An) =

=^ ' Г - - ' А - 1 ' А + Г - '

П^

1 Х ··· X ^ A - I X ^ A + I X ··· Х Л ) , if ΛΑ=--Ω,.

The converse proposition is also true under certain assumptions about

{Ωΐ5 SJ (Kolmogorov Theorem); this asserts that a compatible system of

finite-dimensional distributions generates a stochastic process. Here the

role of the probability space Ω is taken by the set of all functions with

values in ult and the role of the σ-algebra 8 by the minimal σ-algebra

that contains all the cylindrical sets, that is, the sets of form

{ω: ( (ω), ..., hn(e>))£M

n) MnC^X ...xS,.

We do not lay down precise requirements for the space {Ωΐ5 Si}. In any

case these requirements are satisfied by a linear normed space whose unit

ball is compact in the weak topology and an algebra Si generated by

cylindrical sets (see [28]).

Events Alt A

2, ..., A

n are called independent if

P(AlA

2...A

n) = P(A

1)P(A

2)...P(A

n).


The random variables Elt ..., E

n are independent if for any choice of

#i, . .., Bncz2i the events A^ = {ω: £(ω) e Bk\ (k = 1 n) are

independent. In other words, random variables are independent if the

measure in the product space generated by the aggregate of these variables

is the product of the measures generated by each of the random variables

separately.

If £i and B,2 are independent random variables, then the probability

distribution for their sum is easily calculated from their probability

distributions ΜΊ and |i2; it is

μί(Μ-ζ)μ

2(άχ). (2.11)

Below we consider stochastic processes in spaces ,£ja. In these arguments

the socalled Wiener process, which we shall construct immediately, plays an

important role. This process w(t) (to 4. i < To) must have the following

properties:1

a) it is a process with independent increments, that is, for

ίο < *i < t2 < t

3 < i4.< T

o the random variables w(t

2) - w(ii),

w(tA) - w(t

5) are independent;

b) the random variable w(t2) ~ w(t\) (to 4- ii < t

2 < T

o) has a Gaussian

distribution with mean zero and correlation operator I(t2 - ti)·

Since w(t3) - w(t

1) = w(t

3) - w(t

2) + w(t

2) - u>(ti), (2.11) implies

that

,-/ι) (Μ) =-- ^ μΓ(ί2-/

1) (Μ-χ) μ/

(/3-/

2) (dx), (2.12)

Ωι

and w(t) must necessarily have distributions satisfying this. Lemma 2.1

implies that the probability distribution [ijt can be realized in 4?-ι· We

can verify that in this case the relations (2.12) hold.

Using the independence of the increments it is not hard to construct

the system of finite-dimensional distributions of the process w(t) (more

precisely, of w(t) -w(to))- It then follows from (2.12) that this system

obeys the compatibility conditions. By the Kolmogorov Theorem we deduce

that the process w(t) can be realized in the space of functions with values

in £)_!. In the following we shall meet integrals of vector functions with

values in certain spaces. These spaces are always reflexive, and hence the

integrals can be understood in the weak sense.

The integral of a random variable £ over a probability space

Ml -= Ι (ω) Ρ (άω)h

is called the mean value of this random variable. By the map χ = £(ω) the

quantity Mt can be put in the form of the integral

Ml -= [ χμ (dx)

over the space in which the values of £(ω) lie.

1 We should write w(t, ω). Here and below the variable ω is often omitted.

24 Yu.L. Daletskii

Let 3* be a subalgebra of £. The random variable £(ω) is, in general,

not measurable with respect to jF. However, there is another random variable

£ι(ω) that is measurable with respect to SF, is defined up to a set of

measure zero and has the property that

\ Ι, (ω) Ρ {dm) = \ Ι (ω) Ρ (άω)

м м

for every M^j^. This random variable is called the conditional mean of

B, with respect to jF and is written as M(|[jF)·

If ер is a minimal algebra with respect to which all the random

variables £α(α e Л) of some family are measurable, this is called the con-

ditional means with respect to this family and is written M(B,\e,a(oi e A)).

f 1 for ω eM,Let Μ ζ 2 and Х м И = |

0 f Q r щ^ The quantity Μ (χ

Μ \ &)= Ρ (Μ \ &)

is called the conditional probability of the event M' with respect to Si-

Let us consider a stochastic process S,t(t

04 t < T

o) with values in

This process is called a Markov process if for any s and t (to<s< t<To)

the conditional distribution of B,t with respect to the family of random

variables Ят (τ < s) coincides with the conditional distribution with re-

spect to B,s:

The quantity

is called the transition probability of the Markov process. The transition

probabilities satisfy the Chapman-Kolmogorov integral equation

P(s, x, t, M)=\ P(T,y;t, M)P(s,x-T,dy) (s<x<i) (2.13)

Si

and the condition P(s, x\ s, M) - Хм(х).

2. Stochastic integrals. By means of the stochastic integral of Ito

we can construct considerably more general types of stochastic processes

beginning with the Wiener process.

Let w(t) be a Wiener process with values in JQ_±, as described in the pre-

vious subsection, defined on a probability space {Ω, £} (for Ω we can take

the space of functions with values in §_j, as we have seen, but is more

convenient not to consider the structure of Ω). Let jF/ (to< t < T) be the

subalgebra of S generated by the collection of random variables

u>(t) (to< 14 t) and, possibly, some set of events $0 independent of the

process JFf.

Thus, the following conditions hold:

a) &на^

и_ (?ο<*ι<Λί<Γο);

b) the random variable w(t) is measurable with respect to Jft\

c) the random variable w(t + τ) - w(t) does not depend on the events

of the algebra 3F t (τ>0).


Let К denote the class of vector functions /(ω, t) (ω e Ω, t € [to, To])measurable with respect to the set of variables and satisfying the follow-ing conditions:

a) /(ω, /)££ (ω £ Ω, to<t^T

o);

b) /(ω, ί) is measurable with respect to jft;

c) \ Μ || /(ω, t)\\2dt<: oo.

Let /f-y be a class of operator functions satisfying the conditions:

a) Л (ω, /)£{£->&,}, Γ Μ (ω, /)£@2;

b) Л(ω, t) is measurable with respect to jWt\To

c) \ Μσ:;|Γ\4(ω, /)]d/<oo.'o

Note that if condition a) is replaced by the strongera') Λ (со, 0£{U-i->£vL

then condition c) follows from the inequality

σο (TyA) - σ

2 (PAT• T

1) < |j Τ

λ AT \\ σ

2 (Γ""

1) and the condition

To :r0

с') \ Μ !| Л (ω, /) || 11, ν dt - \ Μ | ГУЛ (ω, г) Г ||

2 dt < оо.

'Ό ίο

We denote by the Κγ class of operator functions that satisfy the condi-

tions a'), b), c'). Let φ^ (k = 1,2,...) be the set of eigenvectors of T,

orthonormal in £>: Tq>k = ^k9k- We consider the expansion

w(t)= Σ wk{t)y

k. (2.14)

The coefficients wk(t) =•(w(t), φ&) are mutually independent scalar Wiener

processes. For

Μ [wk (t

2) - w

k (t,)}

2 = M(w (/,) - w (ti), φ

Α)

2 - (φ

Α, φ

Α) (U - t,) = t

2 — t

x,

Μ {[wh (h)-w

k (u)] [wj (U)-wj (t

3)]} =

= Μ {(w (t2)-w(t

i), φ

Α) (w (U) - w (/

3), φ,)} = 0 {к Ф j).

Now let /(ω, t) e К. The expansion00

/ (o), 0 - Σ fk (ω, Ο φ Α ι /ft = (/, <pft), (2.15)

converges strongly in § almost everywhere with respect to ω. it also con-

verges in mean, and

Го оо To

\ Μ (I / (ω, 0 I,2 dt == V \ Μ | f

h (ω, t) |

2 ί//. (2.16)

/ο Λ = 1 ίο

26 Yu.L. Daletskii

The functions /^(ω, ί) and (ω, t) have the same properties with re-

spect to $pt as /(ω, ί) and w(u, t). This allows us to consider the scalar

stochastic integrals

To

fk(t)dw

k(t). (2.17)

We recall (for details see [δ], [2l]) that these integrals are con-structed in the first place for the step functions

fit) = f (tj 4 t < tj + l t to < ti ... < tn < To) by means of the formula

To

t0 j=0

and then in the general case by a limiting process in mean.

Here we have

To T0

Ό

To

ίο

(2.18)ίο

From the independence of the processes

(2.17) are mutually uncorrelated:

it follows that the integrals

To T0

ίο ίο

We now put, by definition,

To oo To

/ (/) = [ (dw (o, / (0) = 2 \ h (odw* (0·

t(i ft=l ίο

(2.19)

This definition has a meaning, because the series on the right converges in

mean. For

n+p To n+p To

Μ 2 \k=n to

(2.20)Ik=n to

and the expression on the right tends to zero as η -» oo by (2.16).

The integral I(f) has the following properties:

a) MI {f) = 0

This equation follows immediately from (2.18) and (2.19).

To

b) M\I(f)\*=[ M\\f(t)\fdt.

ίο

For it is enough to put η = 1 and ρ -» oo in (2.20)

c) If f{t) = Я (tj4· t < tj4 i. *o < ti < ... < in < T

o), then

/(/)= Σ (f\w(tj+l)-w(tj)).

•j=o

(2.21)


To prove this it is sufficient to use the expression for the stochastic

integral of scalar step functions.

We mention that the expression (2.21) has a meaning, since the compon-

ents of the type (/, B,) for / £ £> are measurable linear functions on ,£>_!,

although they are not continuous functionals with respect to ξ£.*ϋ-ι·

Properties b) and c) show that the integral /(/) can be defined in the

usual way, beginning with step functions by (2.21) and then by passage to

the limit.

Now let Л(ω, t) £ Ky. We define the integral

To

I{A)~\ A(t)dw(t), (2.22)

ίο

by putting for any φ £ <y-v

This definition is correct, because /4*φζ <g if Л £ Ky.

The properties of the integral I(A) are easily deduced from those of

(2.23)

a)

b)

In

and so

M| | / (

Ml (;

M\ I

fact,

J (

Λ) R<

l)=0;

(Λ) iv

1)|p-,

У М I

Го

\

<o

ίο

Л 7 (Л)

(/1*7'γ(ί

oo

м2ft==l

(Г

| 2 =

h)

ik

'Л

/i

1*7

(0)

oo

— i

oo

у

Ty

To

\

ίο

| | 2

Μ

dt

1),

—

4>h)2 -^

To

to

y.h=i

fdl

{Α*Ί

I (A*Tyyk)

ToiY) dt=\M

'ίο

If Л б /(у, then (2.23) implies the inequality

To

(Α)\\1<σΙ(Τ-ΐ) \ M|M(/ (2.23')

We can define I (A) not as a weak integral, but just as above for !(/),

beginning with step functions and then by a limiting process in the sense

of strong convergence in mean relative to ω.

The class of functions /(ω, t) for which the stochastic integral I(f) is

defined can be extended. Let K' be a. class of functions

/(ω, t) (ω ε Ω, ί £ [to, Γο]) with values in ,@, measurable with respect

to Ft and such that the integral

To

II / (ω, O i l 2

28 Yu.L. Daletskii

exists for almost all ω. The stochastic integral I(f) for functions of

class K' can be defined by a limiting process from step functions. This

integral has the following property. Ifτ

lim ? || /n (CD, ί)-/(ω, /)|j

2^ = 0

in the sense of convergence in measure, then I(fn) -* I(f) in the same sense.

The proof of these facts is the same as that in [δ] for the scalar case.We now consider the stochastic integral

t

Ι (ω, t) = \ Α (τ) dw (τ)

о

with variable upper limit. It is a stochastic process, which can be investi-gated by methods which hardly differ from those in the finite-dimensionalcase. In particular, the process £(t) has a separable modification withvalues in some compactification of <Q

y whose trajectories are almost all

strongly continuous. In what follows we always suppose that £(t) is chosen

in this manner. Then we need not turn our attention to the possibility that

the trajectory £(t) goes outside ξ)ν, since £(t) remains in $g

y with a

probability equal to unity.

In the following we investigate a more general type of stochastic pro-

cess:t t

Ι (ω, Ζ) = ξ0 (ω) 4- α (ω, τ) άτ + \ Λ (ω, τ) dw (τ), (2.24)

where Λ (ω, τ) ζ Κν, α (ω, τ) ζ ^ ν is measurable for each т with respect toFx, and

Го

J Μ || α (ω, τ)||»<7τ<οο.ίο

The formula (2.24) can be written briefly as dt = a(t)dt + A(t)dw(t)and the expression d£ is called a stochastic differential.

L E M M A 2.2. Let Λ(ω, t) e Ky, and suppose that f(t, x) has for

x ζ §γ> t €. [to, To] continuous derivatives f't, f'x, f'xx, The stochastic pro-cess T\(t) = f(t, £(t)) has a stochastic differential given by the formulaof ItS

di\ = [f't(t, |(0) + И0./i(«. £ (0)) + γδΡ^*(0/**('. t(t))A*(t)]dt +

+ (A*(t)f'x{t,l(t)),dw{t)). (2.25)

PROOF. Let Pfe(fe = 1,2,...) be a complete sequence of projectors of theclass βό· t n a t commute with T. We consider the stochastic processwith the stochastic differential

dlh (t) - Pha (I) dt + PkA (t) Ph dw (i), lk0 = Ρ Α ξ 0 .


This process is finite-dimensional. For such processes the formula of Itois known and can be written in the form

t

/(/, tk(t))-f(t

o,tko)= Ь/Ητ, lk(T)) + (Pka(r), /1-(τ, l

h (τ))) +

to

+ у Sp PhA* (τ) P

hf"

xx (τ, £

ft (τ)) P^P,] d% +

(dw (0, PhA*P

hf'

x (τ, ξ

Α (τ))). (2.25')

ίο

To prove the lemma we must carry out a limiting process as Pk -* I inthis formula. We suppose to begin with that all the derivatives of f(t, x)are bounded.

We observe that TyA (τ) Τ = Β (τ) £ <Β

0 and by Lemma 1.1

lim σ2 [Т^Л — Г

УР

АЛР

А] = lim σ

2 [(5 — P

kBP

h) Τ'

1] = 0.

fe->0O ft-УОО

Hence

to

ι

3 j[ Μϋ\ [ТУА (τ) - ГУ/>АЛ (τ) Pk] dx —> 0 (к - > оо),

and so the process £(t) is approximated in mean by the processesUsing the boundedness of the derivative f'

x(t, x) we find that in this case

UmM\f(t, l(t))-f(t, ξ*(0)|2 = 0,

fc->oo

that is, the limit process in the left-hand side of (2.25') is valid. It isnot hard to see that the limit process on the right-hand side of thisformula is also valid. This fact is evident for the first two components,and for the fourth it follows from the estimate of the stochastic integral,because

II PhA* (τ) P

hf

x (τ, h (τ)) - Α* (τ) f'

x (τ, ξ (τ)) || <

•<\\PkA*(T)Pk\\su

V\\f

xx\\-\\l-l

k\\

y + \\(P

kA*P

k-A*)f

x\\.

Finally, to estimate the third component we have to use Lemma 1.4 andconsider in place of P^ a subsequence for which £&·(*, ω) converges almosteverywhere.

To prove the lemma in the general case it is enough to carry out a newlimit process, approximating f(t, x) together with its derivatives by asequence of functions f

n(t, x) that have bounded derivatives, so that the

convergence is uniform on compact sets. Since almost all the trajectories£(i) are continuous and hence compact, the limiting process in (2.25) canbe carried out in the sense of convergence in probability.

Using (2.25) we can estimate the leading moments of stochastic integrals

30 Yu.L. Daletskii

(this approach was proposed by Skorokhod, an analogous approach was given

by De Dyu Gen in a lecture at the International Congress of Mathematicians

in Moscow). Let Λ(ω, t) e Ky and let

To

Moim[T

yA(t)]dt<:oo.

Then

To

A(x)dw(x)2m _ , ο Γ - ι jj Mof[TyA{t)]dL (2.26)

to

To prove this we apply the Ito formula (2.25) to the case

t

l(t) = i Α (τ) dw(x)

to

and f(x) = I! я;|| γ™. An easy calculation shows that

(f"(r)h h)~m\

and therefore

From (2.25)

= m ( 2 m - 1) || χ ||,2 {m~[) o\ (ТЧ).

(τ)/; (τ, ξ (τ)),

Since the second component has zero mean,

Μ | | I (0 ! | f = {M[SVA* (τ) f"xx (τ, ξ (τ)) .4 (τ) dx.

to

This inequality proves that Λ/||£(ί)||γ is monotonic, because the calcula-tions above show that БрЛ*/"Л>0.

Using Holder's inequality we obtain

To

M||i(7O)||5m<

m(2m —2

{t)\\^m-ô\{TyA{t))dt

m-l To

l\M\\l(t) \fy

m dt] "l -Ц МоТ (ТЧ (τ)) dx]m

to to

_ m (2m — 1)m — 1 T 0

(To-10) m \M\\l{T0)\\lm}


from which after division and exponentiation we obtain (2.26).

3. Stochastic equations. Let a (t, χ) ζ <§Y and A(t, x)£{^—>$

Y) be

functions measurable with respect to the pair of variables / £ [t0, T

o], χζ$£.

The purpose of this subsection is to consider stochastic differential

equations

<2ξ(ω, t) = a(t, ξ (ω, t))dt+A(t, ξ (ω, t)) dw (ω, t). (2.27)

It is convenient to begin by studying, under certain assumptions about

the coefficients, the properties of the integral operator

t t

(SI) (t) = φ (0 + J α (τ, ξ (τ)) dx + J Α (τ, ξ (τ)) άω (χ). (2.28)ίο ίο

This operator can conveniently be considered in the space 2Ιγ offunctions t(t, ω) (ω £ Ω, t

0 < t < To), JF* measurable and taking values in

$Qy Then Щ is a Banach space relative to the norm defined by

((ξ))ί(= sup Μ||ξ(ί)||γ.

Let f (t, x) (t£[t0, T

o], x(z$Q

y) be a function with values in a normed

space. We say that it satisfies condition L if the following inequalities

are valid in the corresponding norm for all / ζ [t0, T

o], Χχ, г€^т :

||/(г, х)\\<Сх + С2\\х\\у,

Ш/ Ύ \ { (t v \\\ <^ Г1 \\ г 'г IIt 1/ / \ ? 2 / 1 1 ^ > ^ 2 *^1 *^2 V*

Below we always suppose the function Φ(ί) to be independent of u>(O·The algebra Jft involved in the definition of the stochastic integral can

be taken to be generated by the random variables

ιυ(ω, χ) (/0<τ</), φ (ω, τ) ( ί

ο< τ < ^

ο) .

LEMMA 2.3. Let φ^Ηγ™ and let the functions a(t, χ), TyA(t, x)satisfy condition L in the spaces £jv and &2($Q), respectively. Then the

operator S acts in the space Шу and is continuous in it.PROOF. I t is c lear that (SB,) (t) i s measurable with respect to the σ-

algebra Jft.Further, from the inequa l i t i e s

t

Μ || (SI) (I) - φ (Ζ) ||Y2m < C3 [ Μ || α (τ, ξ (τ)) ||f" dx +

- 1((-3-~Cb)[Ci (T0 — !0)-]

rC2 \ Μ Ι | | ( τ ) | | Υ dx],

it follows that for ££Иу"1 a l s o 5ξ

32 Yu.L. Daletskii

The analogous estimate

;2 - SlJf™ = sup Μ || (Sl2) (t) - (SI,) (t) \\2

y

m <

t

< sup j C3 [М\\а (τ, ξ2 (τ)) — α (τ, ξ! (τ)) ||?ί ^ J

' ' [ 7 ^ (τ, ξ 2 (τ))-Γ γ ^(τ, ^ (τ))1

?w

(ο

implies the continuity of -S.LEMMA 2.4. Suppose that

a(t,x)£Qy, TyA(t, z ) £ © 2 06v) (*£[*<>» ô). ^ € ^ v )and second order derivatives with respect to χ are uniformly bounded.

Then S, regarded as an operator from to has a continuous

derivative of order k (k = 1,2).

PROOF. We show that the first derivative is given by

We estimate the expression

(2.29)

io

To

It remains to show that each of the components on the right-hand side

of the inequality tends to zero as (X -* 0. Since these components are quite

similar, we consider only the first of them. Since by (1.10)

α (τ, ξ (τ) + ah (τ)) - α (χ, ξ (τ)) _ α'χ (τЛ {τ)) h =

ОС

[αχ (τ, Ι (τ) + α th (τ)) - α

χ (τ, ξ (τ))] h (τ) dt,

we get the estimateTo

\ Μ α (τ, I » α ( τ ' ξ ( ^ 1 _ . α ; ( τ , ξ ( τ ))/, ( τ) tix <

<\dt \Μ{ II α; (τ, ξ (τ) + α th (τ)) - α'χ (τ, ξ (τ)) ||»,

γ || h (τ) ||

2

Υ} άτ,

0 ίο

the right-hand side of which tends to zero by a theorem of Lebesgue.

Infinite-dimensional elliptic operator 33

So we have shown that (2.29) is a weak derivative of S(Z). It is not

hard to check that this expression depends continuously on £. Together with

the boundedness of a'x, A'

x this implies that the weak and strong derivatives

coincide. The continuity of Sg with respect to h is proved just as in the

preceding lemma the continuity of 5 was proved.

The existence of the second derivative, representible in the form

(Ξ'χ (y) y) ---= ξ {a"xxy) ydr+^ (A

xxV) у dw (τ), (2.30)

to to

is shown similarly. However, here the estimates involve the quantity

||/ι(τ)|| which can only be estimated in Щ..L E M M A 2.5. If the conditions of the Lemma 2.3 hold, then for some

η > 1 the operator Sn is a contraction in Щ-:

((Sn (ξ

2) - S

n (ξ,))) < q «h - ξι» (q < 1 )·

PROOF. Integrating the estimate in the proof of Lemma 2.3 we deducethat

2 (To -t

o + i)Cl \ Μ |, (Sy (τ) - (SIO (τ) |j

2 dx

to

and further, by induction,

'1-I ξθ (τ) |,« dx =

It remains to observe that for sufficiently large η

n

3*

Now we can go over to the study of the differential equation (2.27).This equation with the initial condition £(ω, 0) = £

0(ω) is equivalent to

the integral equationt t

t (t) - φ (t) + jj α (τ, t (τ)) c/τ -!- \ Α (τ, ξ (τ)) dw (τ) (2.31)

<Ό /ο

or, what comes to the same thing, the equation <S(i) = S(£)(t) for φ(ί)=£0.

T H E O R E M 2.1. Let φ (0 € Ηγ arid Jet the functions a(f, x)^C\,(^

y),

A(t, x) £ CO i 7 (ig

v) satisfy condition L. Then the stochastic integral equa-

tion (2.31) has a unique solution (up to stochastic equivalence)

34 Yu.L. Daletskii

measurable for each t with respect to JFt and satisfying the conditionsup M\\l(t)\\*<zoo.

lo~i' To

If Φ(ί) is continuous, so is this solution.

PROOF. By Lemma 2. 5 the operator Sn is, for some η > 1, a contraction

in the Banach space Щ.у. By the contraction mapping theorem there is in 2IV

a unique element B, that satisfies the condition SnB, = £. Since for this

Sn(SB.) = Sn + xt = SB, we have SB = B. On the other hand, the last equationimplies that SnB =B, and so В is the unique solution of (2.31) and has allthe properties listed in the conditions of the theorem (these propertiescorrespond to membership of the space 2I

V). The continuity of B(t) follows

automatically from the continuity of the right-hand side of (2.31).REMARK. Quite similarly we can prove by using (2.23) that if φ(0€^ν>

then

sup Λί||ξ(ί)||ί<οο. (2.32)

4. The dependence of the solutions of a stochastic equation on the

initial values. In this and the following subsections we investigate the

manner in which the solutions of the equation

t t

I (t) = L· + Ι α (τ, Ι (τ)) άτ + Α (τ, ξ (τ)) dw (τ) (2.33)ίο <ο

are influenced by variations of its coefficients and initial values.

We can regard the solution t = £(t) (to < t < To) of (2.33) as a func-

tion B, = g(£o) of its initial conditions. This function satisfies identi-

cally the condition Φ(£ο, Β,) = 0, where

/ t

Φ (Ιο, ξ) = ξο + \ α (τ, Ι (τ)) άτ + \ Α (τ, ξ (τ)) dw (τ) - ξ (*). (2.34)

to to

Using the general implicit function theorem and the properties of

φ(£0, Β,) we can establish that g(£

0) is differentiable.

T H E O R E M 2.2. Suppose that the functions a(t, χ) ζ ,ftv,

TyA (t, a;) β

2 ( ) have uniformly bounded derivatives of order k = 1, 2

for t € [to, T o], # € $ v and that they satisfy condition L in these spaces.Then the function B, = g(£o) with values in 9IY defined by ξ0

€ 51 γlias continuous derivatives of order k (k = 1,2).

РЖХЖ. We regard Φ(£ο, £) as a map from Я?* X 2I?

fe to 2Iy\ As a con-

sequence of Lemma 2.4 this function has a continuous first derivative

f t

(<DiA)(*)=jj α; (τ, ξ (τ)) /г (τ) tfx + J A'x (τ, ξ(τ)) Α (τ) dw ( τ ) — Α (τ). (2.35)

/ο <ΌThe equation (Φ^/ι)(ί) = φ(ί) is formally somewhat different from (2.31).However, it is easy to see that Theorem 2.1 can be applied to it, becausethe functions a'

xh, A'

xh satisfy condition L with respect to h in mean.

Consequently this equation has a unique solution h^S&^h for ф£ЭДу*.


I t follows from this that the linear operator (Щ£ {^{yh - · Щ1*} has abounded inverse in the spaces Щ· (k--i, 2).

By the implicit function theorem we can conclude that £ = g(B,0) has asmany continuous derivatives in the space Щ, as Φ(£ο, £)· I t followsfrom Lemma 2.4 that the derivative Φ^ is continuous ?[γ and Φ"ξξ in sjl(,and this proves the theorem.

REMARK. The derivatives of g(£o) satisfy equations obtained by term-by-term differentiation of the relation Φ(£ο, £) = 0:

t t

Δξ0 + \ a'x (g'Alo) dx -\- \ A'x (g Al0) dw (τ) — (g' Δξ0) (t) ----- 0, (2.36)ίο ίο

t t

\ a'x \(g"Al0) Δξ0] dx -f \ A'x {(g"Alo) Δξ0] dw (τ) — \(g"A%0) Δξ0] (0 +ίο to

t t

I \ I /7 ι 0 /\ с j | iff /Д с ϊ ί/ Τ* - ί — \ ί j \ (0 /\ ί 11 ι /3* /\ ? ι /"/)/} ι Τ 1 — ι ) ( Χ "л / ^

ίο ίο

C O R O L L A R Y 2.1. Suppose that f (x) ζ С2 (<g

v) /ias bounded first and

second order derivatives. Under the conditions of the theorem the sameholds for the function (defined with fixed to, t)

u(x)-Mf(g{x)). (2.38)Also

и (x)^Mf'(g(x))g'(x), (2.39)lj" (x) — Μ ( f" (rr (τ)) σ' (χ) σ' (χ)) 4- Μ ί' (Ρ (τ\\ ρ" (τ) (9 ΛΟ\

It is enough to prove these relations in the weak sense. Since theright-hand sides are strongly continuous and locally bounded, the weakderivatives are strong.

We give the proof for the first relation only, because the second isproved quite similarly, though the proof is more involved. First of all weobserve that

2

ι2M\\g'(x-r- ath) — g' (x) \\y, ydt-Ô.

о

36

Further,

м {\ r\gо

Yu.L. Daletskii

(g (x+ah)~g (x))}g ( <

Μ -g'(x) h

Here the second component tends to zero, as proved above, and the second

tends to zero by the theorem by Lebesgue, because the integrand is bounded

and converges to zero in measure.

5. The dependence of the solution of a stochastic equation on the

coefficients. We consider a stochastic equation

ix(s, l(s))ds+[ Ax(s, l(s))dw(s),to t0

(2.41)

whose coefficients depend on a real parameter. We show presently that undersome assumptions on the continuity of the coefficients with respect to theparameter λ the solution ^X(S) of (2.41) also depends continuously on thisparameter.

THEOREM 2.3. Suppose that the coefficientsa%{t, x) € $v> Αχ (t, x) £ { -i—>$3γ} satisfy condition L with constants thatdo not depend on λ, and that the following relations hold:

Km , x)-al0(t, *)

Then

, lim \\\Ax(t, x)-Au(t,λ->λο

, To], х

PROOF. Let

lim supλ-»·λ0 t

be the right-hand side of (2.41). Then

P o r To

To

+ 2

Μ\\αχ{χ, ξ(τ))-α λ 0(τ,

, 1(τ))~Βλ0(τ, ξ (τ))] Γ"1} άτ, where Βλ = Αλ

(2.42)


The expressions | |α λ (τ , ξ (τ)) — αλ ο (τ, ξ (τ)) || andσ2 {Ty [Βχ (τ, ξ (τ)) — Βχ0 (τ, ξ (τ))] Γ'1} are bounded and tend to zero asλ -» λ0 (this follows from Lemma 1.1 for the l a t t e r ) . By the theorem ofLebesgue we can pass to the limit under the integral sign for τ and ω. Byinduction i t i s easy to verify the inequality

ξχο) ~SZ (ô)» < Mm {{Sk (ξλ0) - 5 λ ο (ξλ0))>. (2.43)

Under the conditions of the theorem we can choose an m such that

holds with one and the same constant q < 1 for all λ. Then from

Ik — Ελ0 = £Γ (ξλ) — S%

we get the inequality

and this together with (2.42) and (2.43) proves the theorem.COROLLARY 2.2. Let \1\^ be a measure in the space {igv, 35V}

corresponding to the random variable £\(t) . If the conditions of thetheorem hold, then the sequence of measures \i\t t converges weakly toUXo, t» * n other words V-\ltt * s weakly continuous in the parameter λ.

For if f(x) i s a bounded continuous function on igv, then

\ / (x) μλ0, t (dx) - Mf ( | λ 0 (ω, t)) = lim Jlf/ (ξλ (ω, <)) = И т \ f (χ) μλ, f (dx).^ λ->λο λ->λθ ^

(2.44)

This formula (2.44) is also valid if f(x) i s supposed to satisfy con-dition L instead of the condition of boundedness.

Theorem 2.3 can be used to approximate the solution of the stochasticequation (2.31) by means of solutions of finite-dimensional stochasticequations. Let Pfe (k = 1,2,...) be a (γ, -Incomplete sequence of projectorsof the class *>$'o. We consider the sequence of equations

t t

I (t) = lo. k + \ ah (τ, ξ (τ)) dx+Âh (τ, ξ (τ)) dw (τ), (2.45)ίο <ο

where lo,h = Pklo, ah = Pha, Ak = PkAPk.If the coefficients a and Λ satisfy the conditions of Theorem 2.1, then

so do the coefficients of (2.45). Thus, there is a stochastic process £k(t)that satisfies (2.46). Since the right-hand part of this equation belongsto a finite-dimensional space £ph, the values of the process £&(ΐ) l ie inthis space with a probability equal to 1.

I t i s not hard to see that, if a(x, a;) ( βΥ and Α (τ, χ) ζ {φ-ι —> $ v},the coefficients of (2.45) tend to a, A as k -» oo in the same sense as inthe conditions of Theorem 2.3. This leads to the following result.

THEOREM 2.4. Suppose that the conditions of Theorem 2.1 hold and

38 Yu.L. Daletskii

that Л (τ, χ) £ {£_! —> £ν}. Let P

k (k = 1,2,...) be α (γ, -l)-cojipZete

sequence of projectors of the class Щ'о.

Then the solution £fe(t) of (2.31) can be approximated by solutionsjfe(t) of finite-dimensional stochastic equations:

limsupM | |Ы0-£(0 |1 а = 0·

COROLLARY 2.3. The measure \it in the space {$7, Sv} correspond-ing to a random variable £(t) is a weak limit of the measures \J-k,t corres-ponding to £fc(t), concentrated on finite-dimensional subspaces.

COROLLARY 2.4. A stochastic process £fe(t) that satisfies afinite-dimensional stochastic equation is a Markov process and its transi-tion probabilities are calculated by means of relations

P(t, x, s, Μ)=Ρ{1Η(8)ζΜ\&(ί) = χ} = μ№,.(Μ), where μ ^ ,

s is a

measure corresponding to the random variable !&,*($), satisfying the

stochastic equation

On going to the limit as k -» oo, we obtain the following proposition.The solution of the stochastic equation (2.33) is a Markov process with

transition probabilities P(t, x, s, M) = Ρ {l\x) (s) ξ Μ}, where l\x) (s)

satisfies the equation

s s

l(s)=x+\a(r, 1(τ))άτ+ \ Α (τ, ξ (τ)) Λ* (τ). (2.4.6)t t

6. The absolute continuity of measures corresponding to solutions ofstochastic equations. Let μ and ν be two measures given on the measurablespace {Ω, 2}. We recall that μ i s called absolutely continuous with re-spect to ν if v(M) = 0 ( M £ S ) implies that \L(M) = 0.

If μ i s absolutely continuous with respect to v, then integrals withrespect to μ can be reduced to integrals with respect to v:

5JΩ Ω

provided f(x) i s measurable and at least one of these expressions has ameaning.

Consider the pair of stochastic equations

To T0

lk(t) = lo+ \aik)(r, Η^(τ))^τ+ J Α (τ, lh{x))dw(x) ( A = l , 2). (2.47)

The solution of each of these generates a measure μ in the space of func-tions with values in £jY. Let us examine the conditions under which thesemeasures are absolutely convergent with respect to one another. In the


finite-dimensional case they were obtained in [2l] (see also [29]), whereequivalence of measures corresponding to diffusion processes is studied bymeans of differential equations for distributions). In the case of aHilbert space these results are given in [17], [l9]. Here they are givenin a rather different form.

THEOREM 2.5. Suppose that the coefficients of the equations (2.47)satisfy the condition of Theorem 2.1 and that the operator Л £{£)_!—> ^Y}has an inverse Л" 1 £ {|3γ—>^_!}. If<x(t, x) = A~1(t, x) [a2 (t, x) — «i (t, χ)] £OQ and \\ci(t, x) |j <CCx-\-C2 \\ %\\y,then the measures μ! and \l? are each absolutely convergent with respectto the other and the density can be represented in the form

To Tol o g d j r * [ ^ = - y Ι ϋ α ( τ ' 5i(t))||"dT+jj (α (τ, 5ι (Ό), dw(x)). (2.48)

t'o to

PROOF. Let Pj (j = 1,2,...) be а (γ, -1) complete sequence of operatorsj

of the class ψ0: PjX=^] (x, <Рг)фг· The equation

t t

I it) = Pjlo + I PjaM (τ, ξ (τ)) dx 4- J PjA (τ, Ι (τ)) Pj dw (τ)to to

can be given the form

to г = 1 <о

where wi = (wt q>i), bij = PjAî. Let OLi = ( / 4 " 1 ( a ( 2 ) - q c ( l ) ) , Φι). Then3

The measures generated by this pair of stochastic equations aremutually absolutely continuous and their density has the form

, To j T0 jdV 1

On subst i tut ing the values of uj and v>i we obtain

To T 0

1 С ι С

<0 <0

As j -* oo, t h i s expression tends in mean to

To To

to to

40 Yu.L. Daletskii

Further, by Theorem 2.4 the processes £· tend in mean to £y. By a lenmaof Skorokhod [2l] we can conclude that the measures μ2 and μ± are absolutelycontinuous and the expression (2.48) gives the density of μ2 relative toμι.

7. Examples. Consider the countable system of stochastic equations

t, 1

E*o+ \ ΛΑ(Τ, ξι (τ) , . . ., ξη (τ), . . . ) d x +

+ 2 $ &** (τ, L· (τ), . . ., In (τ), . ..) dws (τ), (2.49)s = l ίο

where ws(s = 1,2,. . .) i s a sequence of pairwise independent Wiener pro-cesses, and afe(T, xlt . . . , ж„, . . . ) , 6fc s(

T. д^ *n, . . . ) are scalarfunctions of a countable number of var iables . We suppose that for somechoice of the numbers λ^, μ^ (k = 1,2,.. .) the conditions

^ΣμϊΊξ*ο|*<οο, Σμ*Ίβ*Ι1<°°. Σ μι^)^Α%<°°

k=l h=l j , k=l

hold. Then

L &\ В = \\band if these functions satisfy condition L in the corresponding norms, thenthe above results apply to (2.49).

We need only note that, while the exposition above is carried out forone scale, as was mentioned in the remark to Lemma 1.4, we can consider apair of scales and suppose that the process W= {u>fc} l ies in one of themand B, = {£fc} in the other. All the arguments go over verbatim.

As another example consider an equation of the form

t

€(«, x) = lo(*)+l α (τ, χ; ξ (τ, ·))<** +'о-

ί

+ ^ J δ(τ, *, χύ ξ (τ, -))LXldw(T, Xl)dxt (2.50)ίο G

in a scale of function spaces.Here the elements of the spaces iQa are ordinary or generalized func-

tions у = y(x). In this context the random variables with values in thesespaces are ordinary or generalized random processes. In (2.50) A i s takenas a differential operator in x.


Chapter I 11

PARABOLIC EQUATIONS OF SECOND ORDER

§1. Equations with constant coefficients

In this section we consider the Cauchy problem for the equation

^ ± * A ) (3.1)

with constant operator 4 { _j -^· §γ}. I t was shown in the first chapterthat the right-hand side of (3.1) makes sense for F (x) £ C2 ($v).

In the following section more general equations with variable coef-ficients are treated. The separate discussion of (3.1) is carried out fortwo reasons. First of all, in this case the Cauchy problem can be studiedfor a wider class of functionals. Besides, in this investigation i t is notnecessary to have recourse to the awkward apparatus of stochastic equations,which we have not been able to avoid in the general case.

THEOREM 3.1. Let Φ(χ) be a functional of the class C2 (igv) whichfor some С > 0, (X > 0 satisfies the condition

ll^-^H?.!} (fc = 0, 1, 2; χζ§,). (3.2)

The Cauchy problem for (3.1) with the condition

F(0, χ) = Φ(χ) (3.3)

has a solution on the interval 0 < t < - that can be represented in the

form

F(t, * ) = J <b{x-y)v.AA*t{dy), (3.4)

where \iAA*t i-s a Gaussian measure in JQy with mean a = 0 and correlationform 5(φ, ψ) = t(A*<p, Λ*ψ),. This solution is unique in the class of func-tionals F(t, x) that satisfy (3.2) and have a derivative F't satisfying thesame inequality.

PROOF. Observe to begin with that the Gaussian measure №A*At i s

countably additive in £)v by Lemma 2.1. I t follows from (3.2) and (2.10)

that the expression (3.4) i s finite for 0 < t < -. Let the projection(X

It was erroneously asserted in the original version that it is enough to supposeφ(χ) to be continuous and satisfying an inequality (3.2) without assuming theexistence of derivatives. I thank Leonard Gross who pointed out to me the in-sufficiency of this condition.

42 Yu.L. Daletski ι

Ρζ.%0 b e s u c h t h a t Φ(Ρχ) also satisfies (3.2). We consider the function

FP (t, x) = J φ (PX - Py) μΑΑη (dy) =

(Рх-у)е 2« dy, ( 3 5 )

where β = A*A. On choosing an orthonormal basis Ψι φη in the finite-dimensional space XP and putting Px = Σχ&φ& we represent Fp as a func-tion of the variables xlt . . . . xn, t satisfying (2.4)

dFP _l_ γ

The last equation can be put in the form

= i- Sp Л*РПРЛ = -i- Sp Л*/1,Л. (3.6)

Now we consider a γ-complete sequence of projectors Pjt (k = 1,2,...)with the above property (we can construct such a sequence by consideringfinite-dimensional projectors "almost commutative" with β). By theLebesgue theorem we can go to the limit in (3.5) and in the equations ob-tained from (3.5) by differentiation under the integral sign. By Lemma 1.5the limiting function, as k -> oc\

F(t, x)= \ <D{x~y)\iAA*t{dy)

has all the necessary derivatives. By Theorem 1.1 we can go to the limitin (3.6), and as a result we conclude that F(t, x) satisfies (3.1).

For the verification of condition (3.3) we consider the difference

F (t, χ)-Φ (χ) = J [Φ (х- у) - Φ (χ)] μΑΑ*ί (dy) =

The first component on the right can be made as small as we please bychoosing a small r, because of the continuity of Φ(χ). The second componenttends to zero for any r, as t -» 0. This follows from

(dy) < t l|rY;f112 Sp [Г 2

which can be proved by arguments similar to those used in the proof ofLemma 2.1.

I t remains to prove the uniqueness of the solution. Let F(t, x) be asolution of (3.1) that has the properties described in the hypotheses andsatisfies F(0, x) = 0. We consider the functional Fk(t, x) = Pfejc, where P


is the sequence of projectors described above. It follows from (3.1) thatFk(t, x) satisfies the equation

"(t, Phx)A)=fSp(A*PkF"k(t, x)PkA) + eh(t, Pkx), (3.7)

where £k(t,Pkx) tends to zero as Pk •* I, by Theorem 1.1, and satisfiescondition (3.2). (3.7) can be regarded as an inhomogeneous parabolic equa-tion in <£рд, and this gives the representation

F(t, Pkx)=\^ ^ еА(т, Ph(x — y))\iAAHt_x)(dy)d%.

Here we can go to the limit as k -» oo by the Lebesgue theorem. Then weobtain F(t, x) = 0, and this implies the uniqueness of the solution.

REMARK 1. In proving the theorem we have proved incidentally that thesolution of the Cauchy problem (3.1), (3.3) can be approximated with anydegree of accuracy by solutions of the finite-dimensional problem (3.6),obtained by means of projection into finite-dimensional space.

REMARK 2. Similarly i t can be shown that the solution of the inhomo-geneous equation

4 f = 4 Sp (A*F"A) + G(z,t),F (0, x) = 0, (3.8)

can be written in the form

t

F(t, x)=^ J G{x-y,x)vAA4t.x){dy). (3.9)

§2. Equations with variable coefficients

I. The inverse Kolmogorov equation for processes defined by stochastic

equations. We consider a differential equation of the form

(3.10)

^FThis equation differs in appearance from (3.1) by the sign before —.

Because of this the Cauchy problem has to have an additional condition atthe right-hand end of the segment

F(T0,x) = O(x). (3.11)

It is natural to consider the inverse equation as well as the directequation in applying stochastic equations. The change of variablet - To - τ turns this into a direct parabolic equation with the usualCauchy problem.

44 Yu.L. Daletskii

The equation (3.10) i s closely connected with the stochastic equation(2.33); the conditional means of functionals of processes £(t) defined bythe (2.33) satisfy (3.10). In the theory of stochastic processes such equa-tions are called inverse Kolmogorov equations.

Let Φ(χ) be a bounded continuous functional on igv. We consider theconditional mean Mx^(B,(s)) of the random variable Ф(Я(«)) subject to thecondition that £(t) has the value x£$Qy at a moment t < s. I t was shownin the previous chapter (see Corollary 2.4) that the corresponding condi-tional probability distribution P(t,x,s,M) i s defined by the measure

(x) (x)

Ut (M) generated by the random variable B,t (s) which satisfies (2.46).

Thus,Mx ίΦ(ξ(«)) = ΜΦ(ξί χ ) (5))= [ Ф(у)Р(1, χ, s, dy)=[ Ф(у)р\х)(ау). (3.12)

THEOREM 3.2. Suppose that the functions a(x,t)(i$Qy,A(x, t) £ {£)_! —-> fey}- are continuous and have continuous uniformly boundedderivatives a'x, axx, A'x, Axx for to < t < To, z £ $ · The Cauchy problem(3.10), (3.11) for the uniformly bounded continuous functional Ф(х) withderivatives ΦΧί Φχχ having the same properties has a solution that isrepresentable by the form (3.12). This solution is unique in the class offunctionals that are uniformly bounded and continuous together with theirderivatives F't, Fx, Fxx.

PROOF. The function F(t, x) = MX, *Φ (l(T0)) (tQ<it^T0, x£$ )has bounded continuous derivatives F'Xt Fx by the Corollary to Theorem 2.2.We show that i t also has a derivative with respect to t and calculate i t .

We remark that for any continuous functional f(x)

Ά/Τ ± (f /t //\\ V 4 (Ί,\ P(f <r f /7j/\ 4 (~\ /4 1 4\•i" x, t \J Vb V II— \ 1 \У) " \τι x i c> аУ)—7 \x)' \o.iO)

Therefore F(t-\-At, x) = MXt t (t + At, ζ(ί)). Moreover, using the Chapman·Kolmogorov formula we can write

,x)=[ Ф(у)Р(1,х,Т0,ау) =

= J P(t, x, t + At,dz)F(t


Now by the I to formula (2.25) we can deduce the relations

-lf[F(t, X)-F{t + Al, x)]=±[Mx,tF

Sp Α* (τ, Ι (τ)) F'xx (t + At, Ε (τ)) Л (τ, ξ (τ)) ] άτ

ί+Λί

The last term falls away, since Mx,t \ (AFX, dw(x)) = 0.

When we take the limit as At -» 0 under the mean value sign, which islegitimate by the Lebesgue theorem, and observe that the expression underthe integral sign is continuous, we can show the existence of a derivativeF't and at the same time obtain (3.10).

(3.11) holds because of the stochastic continuity of the process E(t)

and the equations F (t, χ) = Μ Φ (ξ? (Το)), Φ (χ) = ΜΦ (cf (t)).We now prove the uniqueness of the solution; this i s done along the

same lines as in the proof of Theorem 3.1. If F(t, x) is a solution thatbelongs to the class described in the hypotheses and satisfies F(T0,x) = 0,then the function Fp = F(t, Px) satisfies the following differential equa-tion in the finite-dimensional space XP:

i ^ S P A* С» χ)F" С» Ρ χ ) Α С» χ) + (α (*· χ)* F ('· Ρχ)) =

= ±-SpPA*(t, x)PF"PPAP + (Pa, F'P(t, Px)) + eP(t, Px). (3.14)

The quantity ερ tends to 0 as Ρ -* I uniformly on compact subsets of £jv.For ε = 0 (3.14) i s an inverse Kolmogorov equation for the finite-

dimensional stochastic equationt t

lP{t) = Px-\-\ Pa(t, l(t))dt+[ PA(t,l(t))Pdw(t). (3.15)to to

If [ip is the measure generated by the solution £p(t) of this equation,

then i t follows from (3.14) that

To

ξ , Px-Py)V,{

P

r]{x_1){dy)dT.

I t remains to prove that in this equation we may go to the limit asPk -* I under the integral sign. The sequence of measures \rx' is, by the

46 Yu.L. Daletskii

Corollary to Theorem 2.4, weakly convergent to the measure \it* . Hence(see [δ]) there i s a compact X with the property that μρ^, ί (Φν—Х)<С.Ь(δ an arbitrari ly small number). Since 6pfe tends to 0 uniformly on X andi s majorized outside X by a uniformly integrable function, the passage tothe limit i s possible, and we find that F(t, x) = 0. This proves thetheorem.

REMARK 1. The solution of the Cauchy problem (3.10), (3.11) i sapproximable to any degree of accuracy by solutions of the correspondingfinite-dimensional problems.

REMARK 2. By an immediate calculation we can verify that the solutionof the inhomogeneous equation

under the condition F(T0, x) = 0 i s given by the formula

F(t,x) =

2. The representation of the solutions in the form of integrals over afunction space. Consider a functional / ( * ( · ) ) , defined on a space &y(t0,To)of functions χ(τ) continuous for t 0 < τ <: To and with values in Jgv. Sincealmost a l l the t r a j e c t o r i e s of the process BSX^ (S) are continuous we can

consider the random variable g((*>) = f(£,t* (ω, · ) ) . I t s mean value i s cal-

culated by the formula

\t

x (ω, ·)) = Mx> t f (ξ (•)) = \ / (y)m\x) (dy), (3.16)

Ъу(1, T0)

where mt

x i s a measure in iiy(t, To).We note that if the functional / depends only on a finite number of

values of the function, /(*( ·)) = f(x(î), . . . . *(τη))» then integrationin (3.16) reduces to integration with respect to a finite-dimensionaldistribution of a process

/(*ι, . . . , * η ) Ρ ( ί , a:, Tj, dzOx

Χ Ρ(τ ΐ 5 î, τ2, dx2) · · • ^(τ η -ι, χη-ι, tn. ^ n ) · (3.17)

To calculate the mean of a functional with a less trivial construction

/ ( x ( . ) ) = J ... J F ( τ 1 ? . .., τ η , χ ( τ Ο , ..., χ ( τ η ) ) d T t ... ( 2 τ η ,

where V(Tlf . . . , τ η , x l f . . . , χη) i s a continuous function and integrationi s over the cube to 4. Ту <: To (j = 1,2,... ,n) or one of i t s subsets, we caninvert the order of taking the mean and integrating.


For example,

To To T 0

Μ J dx, J dx2 .. . ξ dxnV ( т ь . . ., τη, ξ!χ ) (τ,), . . . l[x) (τη)) =t τι τη_χ

Το Το Τ0

t Π τη-ι

Χ г (Τι, 3?ι, т 2 , (Ιχ^) . . . г (x n_j, Д п—ι, Τη, dxn). ( ο . Ι ο )

Let the function К(т, дс) and i t s derivativesFT, Fi, F ^ (x^^Y, ί ο < τ < ^ ο ) be uniformly bounded and continuous. Weconsider the functional

To

F (t, x) = Mx, t exp {\V{x,l (τ)) d x } Φ (ξ (Το)) =

To

exp I J F (τ, or (τ)) dx} Φ (χ (Γο)) dm^. (3.19)

Using a method indicated by Kac [30] we show that (3.19) i s the solu-tion of the differential equation

d^ + ^sp[A*(t, x)F"xxA{t, x)] + (a-(t, x), F'x)\V(t, x)F-=0, (3.20)

that satisfies the condition F(T0, x) = Ф(х).Prom the equation

To

exp {jj V(T,

oo To To To

n = l ί

using (3.18) we can derive the representation

F(t,x)= Σ (>„(*,*), (3-21)n=0

where

To

(?0 ( М ) = Л/я, f Φ (ξ (Го)), ρ Λ (/, χ) = J Μ Λ | t F (τ, ξ (τ)) ^ n _ t (τ, ξ (τ)) άτ.

The function

u (/, χ, τ) = Μ*, t V (τ, ξ (τ)) Qn^ (τ, ξ (τ)) (ί < τ),

by Theorem 3.2 satisf ies the equation

- |- + 1 Sp A* (t, x) u"xx A(t,x)-\-(a (t, x), ux) = 0 (t < τ).

48 Yu.L. Daletskii

HenceTo

η* — — Mx, tV {t, Ε (t)) Qn-i (t, Ε (t)) — \ щ (t, χ, τ) dx —

--=V(t, x)Qn-l(t, x) + y S p i * ( / , x)Qn(t, x)A(t, x) + (a(t, x), Q'n(t, x)),

л7 — ^F Op JCI ^/,, Xj Γ/Q ^t, Xj XX i I, XI —|- ( U Ϊ f, χ ΐ ? Г / д Ц , *С ϊ)

Summing these equations we obtain (3.20) and the equationoo

F (Γο, ζ) - Qo (To, x)+ Σ Qn (Γο, χ) = Φ (χ).n = l

We note further that using Theorem 2.5 we can go from an integral withrespect to the measure mt to an integral with respect to mt , generated

by the stochastic process oSx'{t), which satisfies the stochastic equa-tion

t

1(0 = * + $ Α(τ,%(τ))άω(τ). (3.22)<0

These arguments lead to the following theorem.

T H E O R E M 3.3. Suppose that the hypotheses of Theorem 3.2 hold and

that V(t, x) is a continuous uniformly bounded function having derivatives

V't> V'xt V'xx obeying the same conditions.

The Cauchy problem for the equation (3.20) has a solution representable

by (3.19).

If, in addition A'1 (t, χ) ζ { ^

v- ^ &-i} and ot(t

ix) = A~

1(t,x)a(t,x)îQ

t

then under the hypothesis that \\OL(t, *)||< Cx + С

2||дс||у

To

p1 It γ\ — \ ργ-η J \ I V (Τ Ύ (T\\ I! ft (Τ Ύ (Τ\\ Ι|2 Ι ΑΎ Λ.

To

-+- \ (α(τ, а:(τ)), dw (τ))} Φ(χ(Τ0))πιψ) (dx (·)),

where mt is a measure corresponding to (3.22).

3. Examples. The above results can be used to study the differentialequation of the form

oo oo

j , S=i ft=l

in a function F(t, x) defined on the space of sequencesX ш (Xlt X2, . . . , Xn, . . . ) .


If the function

ak(t,x), %k)\AhJ(t,x)*\

is sufficiently smooth ала does not grow too rapidly as χ increases, thenwe can always choose a scale of spaces in which the corresponding operatorsact. In this way we obtain a class of functions of a countable number ofvariables for which the Cauchy problem i s correct (see Chapter II, §2. 7).

Another example is that of the equation with variational derivatives ofthe form (see Chapter I, §2. 4)

dF (t, y) 1 { f „ , w T

b*F(t, y) , , j_f t = T ) )

G G

The Cauchy problem for such equations is correct in the Sobolev spacesЩ, where k depends on the properties of the coefficients of the equation.

§3. Self-adjoint elliptic operators

Consider the Hubert space ^ v of functions F(*) defined on the spaceand satisfying the condition

ν(άχ), (3.24)

where |iy i s a Gaussian measure with zero mean and correlation form

5'ν(φ,ψ) = (7'- 1 < 1 + ν ) φ, Γ- ( ί + Υ 4) .

The scalar product in $Qy is given naturally by

(F,G}= jj F(x)G{x)\iy{dx).

We introduce in ^ v the second order differential operator, startingwith the differential expression of the form

Z2 (F) = Sp [B (x) F' (x)]' (F (χ) ζ σ (£ν)), (3.25)

where B(x) = A(x)A*(x). If Α (χ) ζ {^ -» Ы and Α (χ) ζ {& -> ^ 2 + γ },then since -4*ζ{$-ν—>igi}, 5 (a:) F' (x) € ^2+v and[B (x) F' (x)]' £{$Qy—> $d2+y}· the expression (3.25) is meaningful. We notethat (3.25) can be put in a form considered earlier if we introduce oneauxiliary concept. Let / (χ) = Β (χ) φ, φ ζ ^ _ ν . I t follows from the argu-ment just given that the quantity Sp / '(*) is meaningful. If the operatorB(x) has a continuous derivative Β' (χ) ζ {£)ν—> {$Q-y—> ^2+v }}» then thisexpression is linear in φ, and so there is a β (χ) ζ ^ γ with the same

50 Yu.L. Daletskii

property for which

We put, by definition, β(χ) = Sp B'(x). I t is not hard to verify the equa-tion

Sp [Β (χ) φ (χ)]' = Sp [Β (χ) φ' (χ)] + (Sp Β' (χ), φ (χ)), (3.26)

which shows that

Sp [Б (s) F' (x)]' = Sp 5 (*) Г (x) + (Sp 5 ' (x), /" (*)) =

= Sp A* (x) F" (x) A (x) + (Sp 5 ' (ж), F' (x)). (3.27)

Let Z) denote the set of functionals F(x) that are bounded and con-tinuous together with their derivatives up to the second order inclusive.In fey define an operator SI by putting

%F = Sp [B (x) F' (x)]' - 2 (F' (χ), Β (χ) тт+у) х). (3.28)

The coefficient A(x) can be assumed to be bounded and continuous togetherwith i t s derivatives up to the second order inclusive.

LEMMA 3.1. The operator 51 is symmetric and positive and

\ &*Ff, %*G') μν (dx). (3.29)

PROOF. LetFP(x) = F(Px) and GP(x) = G(Px),

where P(z?$O and l e tPT = TP.

We put Bp(x) = B(Px) and consider the operator in XP

%PF = Sp [BP (x) F' (x)]' - 2 (F' (x), BP (x) T'WMy) x).

If Φι, . . . , Φη i s an orthonormal basis in XP

and χ = 2

then, as is easily verified,

k, s = l


By integration by parts we obtain the formula

, dbsh

dxs дх/ι ' dxs

(2π)2 Up k, s=i

V , ( + v ) i— / I h ftV

Λ ΖΓ — / I h ft

S ^sft "^ I {?p& (XX[ . . ,

2(1+Y)2

(2π)2

= - ξ (BPF'p, G'P)

which turns into (3.29) after a passage to the limit. The remainingassertions of the lemma follow at once from (3.29).

THEOREM 3.4. The operator SI has a self-adjoint closure in H.PROOF. To prove this theorem we have to show that the condition

<(2Ι-/)Φ, F) = 0 (3.30)implies that

F = 0.

I t follows from Theorem 3.2 that there is a semigroup of contractionoperators S(t) with the property that if Φ e D, then the vector function

is also in D and satisfies the equation

Φ (0) = Φ.

We consider the function

a (t) = e~! (S (t) Φ, F)

for Φ e D. It has the following properties:

a) lima(/)=0;

b) a' (t) = e~l (S (t) Φ, F) + e~l (&S (t) Φ, F) = e"' <(И - /) S (t) Ф, F) = 0.

Consequently

a (i) = 0

and i t follows from the equation

(F, Ф) = lim α (f) = 0

52 Yu.L. Daletskii

that

F= 0,

because the set D i s dense in igY.

Received by the Editors August 22, 1966.

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Documents

Infinite-dimensional elliptic operators and parabolic equations connected with them