28.
31.
L . D . Pitt, "A Markov p r o p e r t y for Gaussian p r o c e s s e s with a multidimensional p a r a m e t e r , " Arch. Rat. Mech. Anal., 43, No. 5, 367-391 (1971). L . D . Pitt, "Some problems in the s p e c t r a l theory of stationary p r o c e s s e s on R d, " hadiana Univ. Math. J., 23, No. 4, 343-366 (1973). L . D . Pitt,'-5'Stationary G a u s s i a n - Markov fields on R d with deterministic component," J. Multivar. Anal., 5, No. 3, 300-311 (1975). K. Urbanik, "Generalized stationary p r o c e s s e s of Markovian c h a r a c t e r , " Stud. Math., 22, No. 3,
32.
261-282 (1962). E. Wong, "Homogeneous
29. 30.
STOCHASTIC N.
V.
Gauss-Markov
EVOLUTION Krytov
and
fields," Ann. Math. Stat., 4__00,No. 5, 1625-1634
(1969).
EQUATIONS
B.
L.
Rozovskii
UDC 519.217 519.218
The theory of strong solutions of Ito equations in Banach spaces is expounded. The r e s u l t s of this theory are applied to the investigation of strongly parabolic Ito partial differential equations. INTRODUCTION 1.
Ito
Equations
in B a n a c h
Spaces
The theory of Ito stochastic differential equations is one of the m o s t beautiful and most useful a r e a s of the theory of stochastic p r o c e s s e s . However, until recently the range of investigations in this theory had been, in our view, unjustifiably r e s t r i c t e d : only equations were studied which could, in ana[ogN with the d e t e r m i n i s t i c c a s e , be called o r d i n a r y stochastic equations. The situation has begun to change in the last 10-15 y e a r s . The n e c e s s i t y of considering equations combining the features of partial differential equations and Ito equations has appeared both in the theory of stochastic p r o e e s s e s and in related a r e a s . Such equations have appeared in the statistics of stoehastie p r o c e s s e s (filtration theory of diffusion p r o c e s s e s ) , statistical h y d r o m e c h a n i c s , population genetics, Euclidean field t h e o r y , classical statistical field t h e o r y , and other a r e a s . Concrete examples of equations of this type are presented in the next section. These equations describe the evolution in time of p r o c e s s e s with values in function spaces or, in other w o r d s , r a n dom fields in which one coordinate - the "time" - is distinguished. The object of the p r e s e n t work is to show how to c r e a t e a unified theory which includes both o r d i n a r y Ito equations and a r a t h e r b r o a d class of stochastic partial differential equations. We realize our p r o g r a m by considering equations of Ito type in Banach spaces. sider the equation
More p r e e i s e i y , we con-
du (t, co)=A (u (t, ~o), t, co) dt @ B (lz (t, ~), t, co) dw(t),
(1)
where A ( - , t, co) and B ( - , t, co) are families of unbounded o p e r a t o r s in Banach spaces which depend on the "event" c~ in nonanticipatory fashion, and w(t) is a p r o c e s s with values in some Hilbert space and with independent (in time) i n c r e m e n t s . Such equations are usually called stochastic evolution equations. 2.
Examples
of Stochastic
Evolution
Equations
1. L i n e a r i z e d Equation of Filtration of Diffusion P r o e e s s e s . One of the m o s t important p r o b l e m s of statistically random p r o c e s s e s is the problem of filtration (see [31]). In e s s e n c e , it consists of the following. We consider a two-component p r o c e s s z = (x, y), e . g . , the (n + m)-dimensional diffusion p r o c e s s
d x ( t ) = a (x (t), y (t), t) d t § O (x (0, V (t), t) d w (r d v (t) = g (x (t), V (t), t) dt + ~ (V (t), t) d w (t), x (0) = Xo, v (0) = v~, 1979.
T r a n s l a t e d f r o m Itogi Naukt i Tekhniki, Seriya Sovremennye P r o b l e m y Matematiki, Vol, 14, pp, 71-146,
0090-4104/81/1604-1233 $07.50 9 1981 Plenum Publishing Corporation
1233
where d i m x = n, and w(t) is the standard m - d i m e n s i o n a l Wiener p r o c e s s . It is a s s u m e d that the component x of the p r o e e s s z is nonobservabte. It is required to find the best m e a n - s q u a r e estimate of f(x(t)), where f is a function known on the b a s i s of observations of the t r a j e c t o r y of the observable component y up to time t. In other words, this estimate is to be sought as a functional of the t r a j e c t o r i e s of the component y up to time t. It is well !mown that such an estimate is the conditional m a t h e matieal expectation of f(x(t)), relative to the ~-algebra generated by the values of y up to time t, i . e . , M[f(x(t)) [5"d]. The filtration p r o b l e m eonsists in computing this conditional mathematieal expectation. In [27] we sueceeded in showing that under b r o a d assumptions 34 [f (x (/))t~~vt] = o ~ f (x) %(x) d x ( I % ( x ) d x t-1, \ Rd
Rd
(2)
,~
where qot(x) is the solution of the Cauchy problem d% (x) ~ {~ tr D ~ (bb*% (x)) -- D~ (a% (x))} dt q- [(~*) -~/2 g% (x) + D~: ((aa*) -~12ob*% (x)] (ov*)-~/2dy (f),~
% (x) - P (xoedx)/dx, Dxx is the m a t r i x of second derivatives, and Dx is the v e c t o r of first derivatives. differential equation with unbounded o p e r a t o r s of "drift" and "diffusion."
This is a linear stochastic
2. Equations of Population Genetics. One of the most important types of models of population genetics is the model with geographie s t r u c t u r e . These are models in which the s t r u c t u r e of the population changes not only in time but also in space (geographically). Various probabilistic models of this sort have been p r o posed by Bailey [41], Crow and K i m u r a [521, MM6cot [72], and o t h e r s . All these models are discrete. Dawson [56] and Fleming [60] proposed continuous (in time and space) models which are limits of the diserete models mentioned. These works of Dawson and Fleming continue conceptually the welt-known work of FeLler [501. The equation proposed by Dawson for the m a s s distribution of the population p(t, x) has the form
dp (t, x)=aAp (t, x) dt + c Iz'-p-(t, x) dw (t, x),
(3)
while the equation of Fleming has the form
dp (t, x) ~ {Ap (t, x) @ ap (t, x) -- ~} dt + V
p (t, x) (12--P (t, x))+ dw (t, x).
(4)
In both c a s e s A is the Laplace operator, a , /3, c are constants, ( a ) . = a V 0 , and w(t, x) is a Wiener p r o c e s s with values in L2(Rd) (d = dimx) and n u c l e a r (see, e . g . , [17]) covariance o p e r a t o r . This means that w(t, x) is a stochastic p r o c e s s with values in L2(Rd), such that for any function e ~L2(R d) We
(t)
f w (t, x) e (x) dx G/
Rd
is a o n e - d i m e n s i o n a l Wiener p r o c e s s and M (we, (t)-- We, (s)) (w~ (t) -- We~(s)) = (t-- s) elQe2, where Q is a n u c l e a r o p e r a t o r on L2(Rd), and elQe 2 is the quadratic form it g e n e r a t e s . Wiener p r o c e s s e s with values in Hilbert spaces are d i s c u s s e d in m o r e detail in Chap. 1, See. 2. 3. System of N a v i e r - S t o k e s Equations with Random External F o r c e s . In the physies literature on the t h e o r y of turbulence (see, e . g . , Novikov [35], Monin and Yaglom [33], Klyatskin [24] and the literature cited there) a model of the motion of an i n c o m p r e s s i b l e fluid is c o n s i d e r e d under the action of random external f o r c e s ; the model is d e s c r i b e d by the following s y s t e m of N a v i e r - S t o k e s equations:
dai(t,x)=
vAu~(t,x)--XUkOu~(t'x) k=l
Oxk
Op dt@dw~(t,x),
c)xi
3
~__~ Oak ~0. k=l
dx~
Here * is the symbol for the conjugate; the a r g u m e n t s x, t, and y(t) of the coefficients have been dropped.
1234
(5)
H e r e u = (u 1, u2,u3) is t h e v e l o c i t y v e c t o r ; p, p r e s s u r e ; v, v i s c o s i t y ; and w i ( t , x), i n d e p e n d e n t W i e n e r p r o c e s s e s with v a l u e s in function s p a c e s . F o r Eq. (5) in a c y l i n d e r (0, T) x G, w h e r e G i s a d o m a i n in R ~ with b o u n d a r y F , the f i r s t b o u n d a r y - v a l u e p r o b l e m h a s b e e n c o n s i d e r e d : tt (t, x ) [ 1 0 . r l x r = 0 , u(0,
x)=Uo(X).
4. E q u a t i o n of the F r e e F i e l d . L e t ~(R ~+~) b e the s p a c e of r a p i d l y d e c r e a s i n g f u n c t i o n s on R d+l, and let g' b e the dual s p a c e of S c h w a r t z of s l o w l y i n c r e a s i n g g e n e r a l i z e d f u n c t i o n s . We d e n o t e b y ~ the ~ - a l g e b r a in g' g e n e r a t e d by c y l i n d e r s e t s . On the s p a c e (g~, ~) it is p o s s i b l e to c o n s t r u c t a p r o b a b i l i t y m e a s u r e v with characteristic functional
C~(~)= t e~v(do~)=exp {--(~l, 1 d
w h e r e ~]E~, At,~=
02
+
02
ot---~,m i s a n u m b e r , and ~co is the v a l u e of the f u n c t i o n a l w o n ~]~.
It is known ( s e e , e . g . , the m o n o g r a p h of S i m o n [37]) t h a t the f r e e f i e l d is one of the s i m p l e s t o b j e c t s of r e l a t i v i s t i c q u a n t u m m e c h a n i c s ; in the E u c l i d e a n m o d e l it can b e i n t e r p r e t e d a s a c a n o n i c a l , g e n e r a l i z e d r a n d o m f i e l d [ i . e . , ~ (co, t , x) ~- co(t, x) f o r eachco~g'] on the p r o b a b i l i t y s p a c e (8', ~;, ~). F u r t h e r , let ~ b e g e n e r a l i z e d white n o i s e , i . e . , the c a n o n i c a l , g e n e r a l i z e d r a n d o m f i e l d on the p r o b a b i t i t y s p a c e (g', ~, p) w h e r e g is the G a u s s i a n m e a s u r e w i t h c h a r a c t e r i s t i c o p e r a t o r
(see, e . g . , [17]). Hida and Strett showed (see [64]) that the Euclidean free field ~ (t, x) is a solution stationary in t of the equation o~ (t, x) ot
o
1 / - ~ + m~'~Ct, x)+w(t, x),
d
where
&x=~=~
Ox~ is u n d e r s t o o d in the s e n s e of the t h e o r y of g e n e r a l i z e d f u n c t i o n s .
Regarding this equation,
s e e a l s o the s u r v e y of D a w s o n [57]. The w o r k of A l b e v e r i o and t t o e g h - K r o h n [40] is a good e x a m p l e of the use of s t o c h a s t i c e v o l u t i o n e q u a t i o n s in E u c l i d e a n f i e l d t h e o r y . years. rigor.
The e x a m p l e s p r e s e n t e d f o r m a s l i g h t p a r t of the s t o c h a s t i c e v o l u t i o n e q u a t i o n s c o n s i d e r e d in r e c e n t We have s e l e c t e d t h e s e e x a m p l e s , s i n c e t h e y have b e e n s t u d i e d in d e t a i l at a m a t h e m a t i c a l l e v e l of
M o d e r n p h y s i c s j o u r n a l s a r e an i n e x h a u s t i b l e s o u r c e of s t o c h a s t i c e v o l u t i o n e q u a t i o n s of the m o s t v a r i e d type w h i c h a r e s t u d i e d only a t a p h y s i c a l l e v e l of r i g o r . 3. and
Stochastic Linear
Evolution Stochastic
Equations Evolution
with
Bounded
Coefficients
Equations
The i m p e t u s f o r the f i r s t m a t h e m a t i c a l i n v e s t i g a t i o n s in the a r e a of s t o c h a s t i c e v o l u t i o n e q u a t i o n s w e r e n o t , h o w e v e r , t h e d e m a n d s of p h y s i c s o r b i o l o g y b u t r a t h e r the i n n e r r e q u i r e m e n t s of m a t h e m a t i c s , v i z . , of the t h e o r y of d i f f e r e n t i a l e q u a t i o n s with v a r i a t i o n a l d e r i v a t i v e s , in the m i d - s i x t i e s D a l e t s k i i and B a M a n [19, 3, 4] s t u d i e d s t o c h a s t i c e v o l u t i o n e q u a t i o n s w i t h the o b j e c t of c o n s t r u c t i n g a s o l u t i o n of the Cauchy p r o b l e m f o r the K o l m o g o r o v e q u a t i o n in v a r i a t i o n a l d e r i v a t i v e s
OF (x, t) 1 Ot = ~ - t r [B* (x, t)F"(x, t)B(x, t)l+A(x, t)F'(x, t), t~
1235
also bounded and Lipsehitz. This direction was further developed by the authors themselves and their students [20, 7, 8, 5, 61. Moreover, a number of works appeared ([2, 42, 43, 53-55, 74, 78] and others} in which linear stochastic evolution equations were studied for other applications such as the theory of control, filtration, and extrapolation of linear, stochastic, Ito partial differential equations. From the point of view of proving the existence of a solution these works differed little from [3]. It was always assumed that the coefficient A(s) was the infinitesimal generator of a homogeneous or inhomogeneous semigroup T s,t, and that B was bounded and satisfied a Lipschitz condition. We remark that this is not satisfied, generally speaking, for the filtration equations in which A depends on the event. The following equation was considered: t
tt
(t)= To,tuo+ f r~,tB (u (s)) dw (s).
(6)
o
The proof of the e x i s t e n c e and u n i q u e n e s s of a solution of this equation is a c c o m p l i s h e d s i m p l y by the method of c o n t r a c t i o n m a p p i n g s , since Ts, t is a bounded o p e r a t o r . It was then p r o v e d u n d e r additional conditions (or it was not p r o v e d at all) that the solution of Eq. (6) b e l o n g s to the domain of the o p e r a t o r A, and Eq. (1) is equivalent to Eq. (6). It should be noted that the conditions f o r the e q u i v a l e n c e of Eqs. (6) and (1) obtained in t h e s e w o r k s a r e r a t h e r b u r d e n s o m e if they a r e c o n s i d e r e d in a p p l i c a t i o n to s t o c h a s t i c Ito p a r t i a l d i f f e r e n t i a l e q u a t i o n s . H e r e the situation is a l t o g e t h e r analogous to the d e t e r m i n i s t i c c a s e . Indeed, the t h e o r e m s on the s o l v a b i l i t y o f i n h o m o g e n e o u s p a r a b o l i c equations o b t a i n e d , e . g . , by m e t h o d s of p o t e n t i a l t h e o r y (see [39, Chap. 1]) a r e much f i n e r than t h e i r a n a l o g s o b t a i n e d by the t h e o r y of inhomogeneous s e m i g r o u p s for l i n e a r o p e r a t o r equations [23]. Methods of p o t e n t i a l t h e o r y in a p p l i c a t i o n to l i n e a r equations of type (1) w e r e u s e d by Rozovskii and M a r g u l i s [36, 32]. In t h e s e w o r k s it is a s s u m e d that the "diffusion" o p e r a t o r has o r d e r z e r o , while the " p r i n c i p a l p a r t " of the " d r i f t " o p e r a t o r does not depend on the event. In the f i r s t work the unique s o l v a b i l i t y of the Cauehy p r o b l e m is p r o v e d , while in the second a fundamental solution of the equation is c o n s t r u c t e d , and p r e c i s e e s t i m a t e s of it in the H 6 l d e r c l a s s e s a r e o b t a i n e d . By m o d i f y i n g the m e t h o d of [36], Shimizu p r o v e d the unique s o l v a b i l i t y of the s e c o n d b o u n d a r y - v a l u e p r o b l e m f o r the s a m e equation as in [36] (see [87]). It is also n e c e s s a r y to indicate s e v e r a l w o r k s of q u a l i t a t i v e c h a r a c t e r . M a r k u s studied p r o b l e m s of the s t a t i o n a r y p r o p e r t y of solutions of l i n e a r equations [73]; Shimizu p r o v e d some c o m p a r i s o n t h e o r e m s [88]; Mahno studied the p r i n c i p l e of a v e r a g i n g for e q u a t i o n s of type (6) [71]. B e n s o u s s a n [45] u s e d a c o m p l e t e l y d i f f e r e n t idea to c o n s t r u c t solutions of a l i n e a r equation of type (1) f o r B ~ 1. In c o n t r a s t to all the w o r k s c i t e d a b o v e , h e r e it is not a s s u m e d that A is the i n f i n i t e s i m a l g e n e r a t o r of a s e m i g r o u p . This condition was r e p l a c e d by a c o e r e i v i t y condition. B e n s o u s s a n u t i l i z e d the method of d i s c r e t i z a t i o n in t i m e . The c o r r e s p o n d i n g d i s c r e t e equation is e a s i l y s o l v e d due to c o e r c i v e n e s s , and b e c a u s e of this s a m e condition e s t i m a t e s of the a p p r o x i m a t e s o l u t i o n s a r e o b t a i n e d ; a weak l i m i t i n g p r o c e d u r e is then c a r r i e d out. In the d e t e r m i n i s t i c c a s e this m e t h o d was a p p l i e d e a r l i e r by Lions [30]. 4.
Nonlinear
Stochastic
Evolution
Equations
The i m p o r t a n t p r o g r a m begun by B e n s o u s s a n in the w o r k cited above was continued in his joint p a p e r with T e m a m [46]. H e r e the m e t h o d of d i s c r e t i z a t i o n was a p p l i e d to an equation with a n o n l i n e a r "drift" o p e r a t o r . It was a s s u m e d that B ~- 1, while the o p e r a t o r A s a t i s f i e s a m o n o t o n i e i t y condition. Methods of the t h e o r y of monotone o p e r a t o r s f o r m one of the m o s t beautiful a r e a s of m o d e r n n o n l i n e a r a n a l y s i s . The foundations of this t h e o r y w e r e laid in the w o r k s of V a i n b e r g , K a e h u r o v s k i i , Minty, and B r o w d e r [9, 82, 49, 50]. F u r t h e r d e v e l o p m e n t is r e f l e c t e d in the m o n o g r a p h s [30, 48, 51, 15]. A p p l i c a t i o n s of this t h e o r y to n o n l i n e a r e l l i p t i c and p a r a b o l i c equations a r e d e s c r i b e d in d e t a i l in [22]. The work of B e n s o u s s a n and T e m a m g e n e r a l i z e s one of the m e t h o d s of the m o n o g r a p h of Lions [30]. The m e t h o d of m o n o t o n i e i t y in a p p l i c a t i o n to s t o c h a s t i c evolution equations was f u r t h e r d e v e l o p e d in the w o r k s of P a r d o u x [83, 84]. In t h e s e w o r k s a g e n e r a l s t o c h a s t i c evolution equation is c o n s i d e r e d with unbounded n o n l i n e a r o p e r a t o r s of " d r i f t " and "diffusion." The r e s u l t s of P a r d o u x contain the r e s u l t s of [45, 46, 29] as a s p e c i a l c a s e as well as the d e t e r m i n i s t i c s i t u a t i o n d e s c r i b e d by L i o n s [30, Chap. 2, Sec. 1]. A c t u a l l y , they also include the r e s u l t s of [2, 42, 43, 54, 55, 74, 78] as w e l l , although h e r e a r e g i o n w h e r e they a r e f o r m a l l y d i s t i n c t can be i n d i c a t e d . The situation h e r e is analogous to the d i f f e r e n c e between the t h e o r y of d i f f e r e n t i a l equations in d i v e r g e n c e and nondivergence forms. 1236
The s o l u t i o n o b t a i n e d in [84] b e l o n g s to the d o m a i n s of the o p e r a t o r s A and B a n d t s m e a s u r a b l e with r e s p e c t to a ~ r - a t g e b r a c o n s i s t e n t wlth the W i e n e r p r o c e s s and the i n i t i a l c o n d i t i o n . It is c o n s t r u c t e d on a p r e s c r i b e d p r o b a b i l i t y s p a c e ; i . e . , in c o r r e s p o n d e n c e with the t e r m i n o l o g y of o r d i n a r y s t o c h a s t i c e q u a t i o n s , it is a s t r o n g s o l u t i o n . It s h o u l d b e n o t e d t h a t E x a m p l e s 2, 3, and 4 do n o t fit the a s s u m p t i o n s of P a r d o u x . S o l u t i o n s of the e q u a t i o n s d e s c r i b e d in E x a m p l e s 2 and 3 a r e c o n s t r u c t e d in the s e n s e of ( p r o b a b i t i s t i c ) d i s t r i b u t i o n s . The w o r k s [47, 10, 11, 1, 13, 14, 89] a r e d e v o t e d to an i n v e s t i g a t i o n of the N a v i e r - S t o k e s w h i l e E q s . (3) and (4) a r e i n v e s t i g a t e d in [57, 89].
equation,
In t h e s e w o r k s a m e a s u r e is s o u g h t w h i c h is s u p p o r t e d on the t r a j e c t o r i e s of s o l u t i o n s and is e i t h e r a s o l u t i o n of the K o l m o g o r o v o r Hopf e q u a t i o n ( f o r m a l l y ) a s s o c i a t e d with the e q u a t i o n in q u e s t i o n . 5.
Content
and
Organization
of the
Work
In the p r e s e n t w o r k we g e n e r a l i z e the r e s u l t s of P a r d o u x . In c o n s i d e r i n g the s a m e s i t u a t i o n a s c o n s i d e r e d in [84], we a d m i t d e p e n d e n c e of the c o e f f i c i e n t s on the e v e n t (in a n o n a n t i c i p a t o r y w a y of c o u r s e ) . We h a v e s u c c e e d e d in s h o w i n g t h a t c e r t a i n c o n d i t i o n s of P a r d o u x a r e s u p e r f l u o u s ; in p a r t i c u l a r , t h i s is the e a s e of the l o c a l L i p s c h i t z c o n d i t i o n on the " d i f f u s i o n " c o e f f i c i e n t . The M a r k o v p r o p e r t y of the s o l u t i o n (in t) is p r o v e d f o r c o e f f i c i e n t s not d e p e n d i n g on the e v e n t . On the b a s i s of g e n e r a l r e s u l t s r e g a r d i n g the s o l v a b i l i t y of s t o c h a s t i c e v o l u t i o n e q u a t i o n s , we have i n v e s t i g a t e d q u a s i t i n e a r , s t o c h a s t i c Ito p a r t i a l d i f f e r e n t i a l e q u a t i o n s (of any o r d e r ) w h i c h s a t i s f y the s o - c a t t e d c o n d i t i o n of s t r o n g p a r a b o l i c i t y . In the d e t e r m i n i s t i c c a s e (B - 0) t h i s c o n d i t i o n c o i n c i d e s w i t h the w e l t - l m o w n c o n d i t i o n of s t r o n g e t t i p t t c i t y of V i s h i k ( s e e , e . g . , [12]). In the l i n e a r c a s e we i n t r o d u c e d the c o n d i t i o n of s t r o n g p a r a b o t i c i t y in [66] a n d s y s t e m a t i c a l l y i n v e s t i g a t e d it in [26]. An a n a l o g o u s c o n d i t i o n in the l i n e a r s i t u a t i o n w a s i n t r o d u c e d a l s o b y P a r d o u x in [84]. We p o i n t out t h a t in the f i n i t e - d i m e n s i o n a l e a s e o u r r e s u l t s g e n e r a l i z e s o m e w h a t [ [ o ' s c l a s s i c a l t h e o r e m on the s t r o n g s o l v a b i l i t y of s t o c h a s t i c e q u a t i o n s with c o e f f i c i e n t s w h i c h d e p e n d on the e v e n t and s a t i s f y L i p s e h i t z c o n d i t i o n s [23, 18] (see the e x a m p l e of S e e . 2, Chap. II). The w o r k is o r g a n i z e d a s f o l l o w s . A s i d e f r o m the i n t r o d u c t i o n it c o n t a i n s t h r e e c h a p t e r s . The f i r s t c h a p t e r is d e v o t e d to t h e t h e o r y of s t o c h a s t i c i n t e g r a t i o n in H i [ b e r t s p a c e s . H e r e the c o n c e p t of a m a r t i n g a l e and of a W i e n e r p r o c e s s in H i [ b e r t s p a c e i s i n t r o d u c e d , and s t o c h a s t i c i n t e g r a l s o v e r t h e s e p r o c e s s e s a r e d e s c r i b e d (See. 2). In S e e s . 3 and 4 of t h i s c h a p t e r , I t o ' s f o r m u l a f o r the s q u a r e of the n o r m of a s e m i m a r t t n g a l e in a r i g g e d H i t b e r t s p a c e is p r o v e d . T h i s r e s u l t p i a y s an e x t r e m e l y i m p o r t a n t r o t e in the e n t i r e t h e o r y . C h a p t e r II is d e v o t e d to a p r o o f of the m a i n t h e o r e m s on the s o l v a b i l i t y of s t o c h a s t i c e v o l u t i o n e q u a t i o n s and c o n t a i n s an i n v e s t i g a t i o n of the p r o p e r t i e s of t h e s e s o l u t i o n s . In the t h i r d c h a p t e r the r e s u l t s of Chap. H a r e a p p l i e d to i n v e s t i g a t e the s t o c h a s t i c p a r t i a l d i f f e r e n t i a [ e q u a t i o n s of Ito. S e c t i o n s 3 and 4 of Chap. I, w h e r e , a s a l r e a d y m e n t i o n e d , I t o ' s f o r m u l a f o r the s q u a r e of the n o r m of a s e m i m a r t i n g a t e in H i t b e r t s p a c e is c o n s i d e r e d , and S e e . 3 of Chap. II, w h e r e the f i n i t e - d i m e n s i o n a l e a s e is c o n s i d e r e d , m a y b e of i n d e p e n d e n t i n t e r e s t f o r s o m e r e a d e r s . The e x p o s i t i o n in t h e s e s e c t i o n s is t h e r e f o r e i n d e p e n d e n t of the r e m a i n d e r of the w o r k (this i n c l u d e s the n o t a t i o n ) . On the o t h e r h a n d , we go m u c h f u r t h e r h e r e than is r e q u i r e d b y the r e m a i n d e r of t h e t e x t , so t h a t a r e a d e r i n t e r e s t e d in a r a p i d a c q u a i n t a n c e with the i d e a s o f I t o ' s s t o c h a s t i c p a r t i a l d i f f e r e n t i a [ e q u a t i o n s m i g h t r e s t r i c t a t t e n t i o n to a f a m i l i a r i z a t i o n with the a s s e r t i o n s of t h e s e s e c t i o n s . E a c h c h a p t e r is p r o v i d e d w i t h i t s own i n t r o d u c t i o n w h e r e i t s c o n t e n t and o r g a n i z a t i o n a r e i n d i c a t e d . The e n u m e r a t i o n of f o r m u l a s and t h e o r e m s b e l o w i s b i n a r y , and in r e f e r e n c e to a f o r m u l a o r a s s e r t i o n of a n o t h e r c h a p t e r it is t e r n a r y . The l a s t n u m b e r i n d i c a t e s the n u m b e r of the f o r m u l a o r a s s e r t i o n , while the p e n u l t i m a t e i n d i c a t e s the n u m b e r of t h e s e c t i o n . CHAPTER STOCHASTIC 1.
INTEGRATION
l IN
HILBERT
SPACES
Introduction
The t h e o r y of s t o c h a s t i c i n t e g r a t i o n in i n f i n i t e - d i m e n s i o r m t s p a c e s is a r a t h e r b r o a d and r a p i d l y d e v e l o p ing a r e a of the t h e o r y of s t o c h a s t i c p r o c e s s e s . A s u r v e y of the c o n t e m p o r a r y s t a t e of t h i s a r e a is a l a r g e and 1237
d i f f i c u l t t a s k g o i n g f a r b e y o n d the f r a m e w o r k of the p u r p o s e of the p r e s e n t w o r k . In t h i s c h a p t e r we t h e r e f o r e c o n s i d e r only " s e l e c t e d " q u e s t i o n s of the t h e o r y of s t o c h a s t i c i n t e g r a t i o n in H i l b e r t s p a c e s which we u s e d i r e c t l y in the s e q u e l . T h e r e a r e two g r o u p s of " s e l e c t e d " q u e s t i o n s : a) the c o n s t r u c t i o n of a s t o c h a s t i c i n t e g r a l o v e r a s q u a r e - i n t e g r a b l e space ;
m a r t i n g a l e with v a l u e s in a H i l b e r t
b) d e r i v a t i o n of I t o ' s f o r m u l a f o r the s q u a r e of the n o r m of a s e m i m a r t i n g a l e in a r i g g e d H i l b e r t s p a c e . We have c o n s i d e r e d q u e s t i o n s o f g r o u p a) (See. 2 is d e v o t e d to t h e m ) in a v e r y s i m p l e s i t u a t i o n - the integration over a continuous martingale. T h i s s e c t i o n is e n t i r e l y of s u r v e y c h a r a c t e r . It is b a s e d on r e s u l t s o f [67, 81, 78]. The e x p o s i t i o n h e r e is of s y n o p s i s c h a r a c t e r - t h e r e a r e p r a c t i c a l l y no p r o o f s g i v e n . The p r o b l e m f o r m u l a t e d in p a r t b) o c c u p i e s a c e n t r a l s p o t in t h i s c h a p t e r . to i t s s o l u t i o n .
S e c t i o n s 3 and 4 a r e d e v o t e d
It is h e r e a s s u m e d that the H i l b e r t s p a c e H is r i g g e d b y a p a i r of B a n a c h s p a c e s V and V ' [ i . e . , t h e r e a r e the f o l l o w i n g ( d e n s e ) i m b e d d i n g s :V~I-tcV'], and we c o n s i d e r a " s e m i m a r t i n g a l e " of the f o r m f
(t) = i ~' (s) ds + .~ (t), 0
w h e r e v ~ V, v ' ~ V ' , and re(t) is a m a r t i n g a l e in H. I t o ' s f o r m u l a i s d e r i v e d f o r IIv(t)lP~i. The d e r i v a t i o n of I t o ' s f o r m u l a f o r the s q u a r e of the n o r m of a s e r e [ m a r t i n g a l e with a l l c o m p o n e n t s c o n c e n t r a t e d in a s i n g l e H i l b e r t s p a c e d i f f e r s l i t t l e f r o m the f i n i t e - d i m e n s i o n a l c a s e . H o w e v e r , in p a s s i n g f r o m t h i s s i t u a t i o n to that d e s c r i b e d a b o v e , the j u m p in c o m p l e x i t y is c o m p a r a b l e to the j u m p in c o m p l e x i t y in p a s s i n g f r o m b o u n d e d to unbounded o p e r a t o r s . T h i s c o m p a r i s o n is m o r e a p t than is a p p a r e n t at f i r s t g l a n c e . We s h a l l s e e b e l o w that e v o l u t i o n e q u a t i o n s with u n b o u n d e d o p e r a t o r s c a n n o t be c o n s i d e r e d in a s i n g l e s p a c e - it is n e c e s s a r y to s e p a r a t e the d o m a i n of the o p e r a t o r s V f r o m the r a n g e (V' o r H). T h i s s i t u a t i o n w a s f i r s t c o n s i d e r e d b y P a r d o u x [84]. We have s u c c e e d e d b y an e n t i r e l y d i f f e r e n t m e t h o d in g e n e r a l i z i n g h i s r e s u l t s to the c a s e of n o n r e f l e x i v e and n o n s e p a r a b l e s p a c e s . M o r e o v e r , o u r p r o o f is m u c h shorter. E s s e n t i a l u s e i s m a d e of I t o ' s f o r m u l a f o r the s q u a r e of the n o r m in Chap. II to o b t a i n a p r i o r i e s t i m a t e s of the s o l u t i o n and to p r o v e i t s u n i q u e n e s s and c o n t i n u i t y . To c o n c l u d e the s e c t i o n we g i v e a b r i e f a n d , of c o u r s e , i n c o m p l e t e h i s t o r i c a l of r e s u l t s on the t h e o r y of s t o c h a s t i c i n t e g r a l s in H i t b e r t s p a c e s .
bibliographieal survey
The f i r s t i m p o r t a n t w o r k in t h i s d i r e c t i o n is a p p a r e n t l y the w o r k of D a l e [ s k i [ [19] w h e r e he c o n s t r u c t e d a W i e n e r p r o c e s s with an i d e n t i t y c o v a r i a n c e o p e r a t o r in H i l b e r t s p a c e , o r , m o r e p r e c i s e l y , in a c e r t a i n n u c l e a r e x t e n s i o n of t h i s s p a c e , and d e f i n e d a s t o c h a s t i c i n t e g r a l . L a t e r , on the b a s i s of the w o r k of G r o s s [62, 63], Kuo s t u d i e d a s t o c h a s t i c i n t e g r a l on an a b s t r a c t W i e n e r p r o c e s s in a B a n a c h s p a c e [68]. The r e s u l t s o f D a l e t s k i i w e r e a l s o e x t e n d e d in [6], 69, 75], and o t h e r s . Kunita [67] i n i t i a t e d the s t u d y of the p r o b l e m of i n t e g r a b i l i t y with r e s p e c t to a s q u a r e - i n t e g r a b l e m a r t i n g a l e with v a l u e s in a H i l b e r t s p a c e . M e t i v i e r and his s t u d e n t s ( s e e , e . g . , [74-79]) s u b s e q u e n t l y a c h i e v e d c o n s i d e r a b l e p r o g r e s s in t h i s d i r e c t i o n . The i m p o r t a n t w o r k of M e y e r [81] s h o u l d a l s o be m e n t i o n e d . At p r e s e n t the i n t e r e s t e d r e a d e r can s y s t e m a t i c a l l y study the p r o b l e m on the b a s i s of [80, 81, 76, 77, 68]. 2.
Stochastic
Integrals
in
Hilbert
Spaces
We b e g i n t h i s s e c t i o n by r e c a l l i n g and d i s c u s s i n g s o m e v e r y t r a d i t i o n a l c o n c e p t s and r e s u l t s c o n n e c t e d with m a p p i n g s of m e a s u r a b l e s p a c e s into B a n a c h s p a c e s . L e t (S, E, t~) b e a c o m p l e t e m e a s u r a b l e s p a c e with m e a s u r e , and l e t (X, ~ ) be a B a n a c h s p a c e with the a - a l g e b r a of B o r e [ s e t s ( r e l a t i v e to the s t r o n g t o p o l o g y ) . We denote b y X* the s p a c e dual to X and by xx* the v a l u e of a f u n c t i o n a l in X* on x E X.
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A m a p p i n g ( f u n c t i o n ) x : S - - X is c a l l e d m e a s u r a b l e if f o r e a c h F E ~ {s:x(s)EF}GE. A m a p p i n g x :S - - X is c a l l e d w e a k l y m e a s u r a b l e if for e a c h x* ~ X*, x(s)x* is a m e a s u r a b l e m a p p i n g of S into R t. A m a p p i n g x : S - - X is c a t t e d s t r o n g l y m e a s u r a b l e if t h e r e e x i s t s a s e q u e n c e of m e a s u r a b l e s i m p l e f u n c t i o n s c o n v e r g i n g to x(s) p - a l m o s t s u r e t y .
It is clear that the concept of strong measurability separable.
coincides
with the concept of measurability
if X is
THEOREM 2.1 (Pettis, see [22]). In order that a mapping x :S -- X be strongly measurable, it is necessary and sufficient that it be weakly measurable and there exist a set B~s such that p(B) = 0, while the set of values of x(s) on S\B is separable. In particular, if X is separable, then the concepts of weak and strong measurability coincide. Let (Q, g-, P) be a p r o b a b i l i t y s p a c e with an e x p a n d i n g s y s t e m of o - - a l g e b r a s ~r-t~s a s s u m e that ~-0 has b e e n c o m p l e t e d with r e s p e c t to the m e a s u r e P. D e f i n i t i o n 2.1.
tER-= [0, oo).
We
A r a n d o m v a r i a b l e in X is a m e a s u r a b l e m a p p i n g of (Q, :V-, P) into X.
D e f i n i t i o n 2.2. We say that x(t, co) is an ~ r , - c o n s i s t e n t s t o c h a s t i c p r o c e s s in X if x(t, 9) for e a c h t is a m e a s u r a b l e m a p p i n g of (Q, :~%, P) into X. D e f i n i t i o n 2.3. C o m p l e t e l y m e a s u r a b l e s u b s e t s a r e s u b s e t s of [0, ~) x f2 which a r e e l e m e n t s of the s m a l l e s t ( r - a l g e b r a r e l a t i v e to which all r e a l , J - t - c o n s i s t e n t , r i g h t c o n t i n u o u s p r o c e s s e s without d i s c o n t i n u i t i e s of s e c o n d kind a r e m e a s u r a b l e in (t, co). A m a p p i n g ( p r o c e s s ) x: [0, oo) x fZ ~ X is c a l l e d c o m p l e t e l y m e a s u r a b l e if for any B o r e [ set F ~ X the s e t {(t, co) : x ( t , co} ~ F} is c o m p l e t e l y m e a s u r a b l e . In the w o r k we m u s t s o m e t i m e s c o n s i d e r r a n d o m v a r i a b l e s with v a l u e s in i m b e d d e d B a n a c h s p a c e s . Let V and H be two s e p a r a b l e B a n a c h s p a c e s , w h e r e V is a s u b s e t of H and the o p e r a t o r a s s i g n i n g to an e l e m e n t v ~ V the c o r r e s p o n d i n g e l e m e n t v ~ H is a c o n t i n u o u s o p e r a t o r f r o m V to H. LEMMA 2.1. a) K x is a r a n d o m v a r i a b l e with v a l u e s in V ( r e l a t i v e to the B o r e [ g - a l g e b r a of V), then x is a r a n d o m v a r i a b l e with v a l u e s in H. b) if x is a r a n d o m v a r i a b l e with v a l u e s in H, then {~o :x(~o)~V}~J-. A s s e r t i o n a) of the l e m m a is a c o n s e q u e n c e of the m e a s u r a b i l i t y of a s u p e r p o s i t i o n of m e a s u r a b l e m a p p i n g s . A s s e r t i o n b) follows f r o m the c o m p l e t e n e s s of ~- and the fact that the c o n t i n u o u s i m a g e of a B o r e l (in V) set of V is a n a l y t i c and is h e n c e u n i v e r s a l l y m e a s u r a b l e (cf. [28]) in H. We p r o c e e d to the d e f i n i t i o n of a m a r t i n g a l e with v a l u e s in a r e a l H i l b e r t s p a c e H and of a s t o c h a s t i c i n t e g r a l o v e r a m a r t i n g a l e . We s h a l l c o n s i d e r only m a r t i n g a l e s re(t) which a r e (strongly) c o n t i n u o u s in t and with s e p a r a b l e r a n g e . We t e m p o r a r i l y denote by H 1 the c l o s e d l i n e a r hull of the r a n g e of re(t, co}, t -> 0, a ~t2. ! If h(t, co) • H1, then for a l l (t, co) it is n a t u r a l to set i h(s)dm(s)=O. T h e r e f o r e , in view of the o r t h o g o n a [ d e 0
c o m p o s i t i o n of H into H I and H ~ , it s u f f i c e s to study the i n t e g r a t i o n of f u n c t i o n s with v a l u e s in H 1. T h e s e a r g u m e n t s m a k e the f o l l o w i n g a s s u m p t i o n n a t u r a l , and we adopt it to the e n d of the s e c t i o n : H is a s e p a r a b l e H i l b e r t s p a c e which is i d e n t i f i e d with its dual in the n a t u r a l way. F o r hi, h 2 E H we denote by hlh 2 the s c a l a r p r o d u c t of h~, h2; h~ - h l h l , Ihl[ = (h~)l/2. F o r r a n d o m v a r i a b l e s in H with finite m a t h e m a t i c a l e x p e c t a t i o n of the n o r m it is p o s s i b l e to define the c o n c e p t of c o n d i t i o n a l m a t h e m a t i c a l e x p e c t a t i o n in c o m p l e t e a n a l o g y to the f i n i t e - d i m e n s i o n a l e a s e . This d e f i n i t i o n r e d u c e s s i m p l y to the d e f i n i t i o n of the c o n d i t i o n a l m a t h e m a t i c a l e x p e c t a t i o n in the f i n i t e - d i m e n s i o n a l e a s e . N a m e l y , let G be s o m e s u b - o - - a l g e b r a of : - and tel x be a r a n d o m v a r i a b l e in H with M l x l < ~ . D e f i n i t i o n 2.4. The c o n d i t i o n a l m a t h e m a t i c a l e x p e c t a t i o n of x r e l a t i v e to G is the r a n d o m v a r i a b l e in H - M[xl G] such that for any y e H
It is c l e a r that the r a n d o m v a r i a b l e so d e f i n e d in unique (a. s.).
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Definition 2.5.
A s t o c h a s t i c p r o c e s s x in H is a m a r t i n g a l e r e l a t i v e to the f a m i l y {:V't} if
a) x is 9 " c - c o n s i s t e n t in H, b) MIx(t)l < ~ f o r all t -> O, c) M i x ( t ) I J - ~ ] = x ( s )
(a.s.) f o r atl s, t >_ 0, s -<- t.
The next t h e o r e m follows i m m e d i a t e l y f r o m Definitions 2.4 and 2.5 and the equivalence of the t h r e e c o n c e p t s of m e a s u r a b i l i t y in s e p a r a b l e s p a c e s . T H E O R E M 2.2. A s t o c h a s t i c p r o c e s s x(t) in H with finite m a t h e m a t i c a l expectation of the n o r m (for e a c h t) is a m a r t i n g a l e r e l a t i v e to the f a m i l y {:V't} if and only if the s t o c h a s t i c p r o c e s s yx(t) is a o n e - d i m e n s i o n a l m a r t i n g a l e r e l a t i v e to this f a m i l y of a - a l g e b r a s f o r each y ~ H. Definition 2.6. As in the f i n i t e - d i m e n s i o n a l c a s e , we say that a s t o c h a s t i c p r o c e s s x(t) in H is a tocai m a r t i n g a l e * and denote it by X ~ o c ( R + , H) if t h e r e e x i s t s a s e q u e n c e of M a r k o v m o m e n t s Tn * ~r (a. s.) such that the p r o c e s s x ( t A ~ ) is a m a r t i n g a l e for e a c h n. We call the sequence {~-n} a localizing s e q u e n c e . F o r s i m p l i c i t y we h e n c e f o r t h c o n s i d e r only (strongly) continuous local m a r t i n g a l e s and m a r t i n g a l e s . The c l a s s of all continuous local m a r t i n g a l e s in H i s s u i n g f r o m z e r o we denote by ~or (R+,/q). It is e a s y to see that the following r e s u l t holds. T H E O R E M 2.3. If xG~or (R+, H), then t h e r e e x i s t s a s e q u e n c e {~-~} of M a r k o v m o m e n t s l o c a l i z i n g x f o r w h i c h M s u p l x ( t / k v ' ~ ) ] 2 < o o . If Mix(t)12 < m f o r any t-> 0, then M s u p l x s]2-<.4MIxt12 for any t-> 0. t>.O
s~.t
The l o c a l i z i n g s e q u e n c e {~-n} is h e n c e f o r t h a s s u m e d to coincide with {rn}. The next t h e o r e m plays an i m p o r t a n t role in the c o n s t r u c t i o n of a s t o c h a s t i c i n t e g r a l o v e r x ~ o ~ THEOREM 2.4.
(R+,)V).
If xC~or (R+, H), then lx(t)l ~ is a local s u b m a r t i n g a l e .
The p r o o f follows f r o m the equality M [I x (t A %) -- x (s A %) ]21Y~l ~ ~ I I x (t A ~.)b ~ lY~I -- I x (s A*~)12 which, in t u r n , follows i m m e d i a t e l y f r o m the m a r t i n g a l e p r o p e r t y f o r the F o u r i e r coefficients of the e x p a n sion of x in a b a s i s of H. Definition 2.7. An i n c r e a s i n g p r o c e s s ( x ) t f o r x~!9l~oo~(R+, H) is c a l l e d an i n c r e a s i n g p r o c e s s f o r Ix(t)l 2 in the D o o b - M e y e r expansion. F r o m the D o o b - M e y e r t h e o r e m it follows that ( x ) t is uniquely defined (a. s.) and continuous in t. As in the f i n i t e - d i m e n s i o n a l e a s e , if x , g~l~or (R+, H ) , then we set ( x , ~ ) t = l {
( x - b y ) t - - ( x - - v ) t}.
It is e a s y to v e r i f y that f o r e a c h t, s ~ 0, t -> s, and e a c h n f o r which the f i r s t a s s e r t i o n of T h e o r e m 2.3 holds f o r the s e q u e n c e ~'n s i m u l t a n e o u s l y f o r x(t) and y(t) we have M I(x ( t A ~ , ) - - x ( s A ~ ) (V ( t A r , ) - - / r (s/\~.))I ~LI = M I ( x, v > ,A~.-- < X, v > ~A,~ I~.1
c~.~.).
T H E O R E M 2.5 ( B u r k h o l d e r Inequality).._. K xEq2~oc(R,, H) and r is a finite (a. s.) M a r k o v m o m e n t , then
+]
214[sup[x(t) l]-.<3Muf ( x ) ~ . A p r o o f of this t h e o r e m can be found, e . g . , in [771. In H we fix an o r t h o n o r m a l b a s i s {h i, i >- 1} and set xi(t) = hix(t). I r i s k n o w n t h a t a l m o s t s u r e t y f o r e a c h i, j the m e a s u r e on the axis [0, ~) g e n e r a t e d by the p r o c e s s (x i, xJ )t is a b s o l u t e l y continuous with r e s p e c t to the m e a s u r e g e n e r a t e d by ( x ) t. F o r f u r t h e r exposition of the t h e o r y of m a r t i n g a l e s , we r e q u i r e the following notation. Let E be a s e p a r a b l e H i l b e r t space which is n a t u r a l l y identified with its dual, let {el} be an o r t h o n o r m a l b a s i s in E, let 2 ( H , E) be the s p a c e of continuous l i n e a r o p e r a t o r s f r o m H to E, and let ~ 2 ( H I E ) b e the s u b s p a c e of 9 H e r e and h e n c e f o r t h we c o n s i d e r m a r t i n g a l e s and local m a r t i n g a l e s only r e l a t i v e to the family /a-d.
1240
ill, E} c o n s i s t i n g of at[ H H b e r t - S c h m i d t with the n o r m
operators.
It is known that -Q~2(H, E) is a s e p a r a b l e H i l b e r t space
w h e r e 11Ell does not depend on the choice of b a s e s in H and E. If Q is a s y m m e t r i c , n o n n e g a t i v e , n u c l e a r o p e r a t o r in N (H, N) we denote by ~Q (H, E) the set of all l i n e a r , g e n e r a l l y unbounded o p e r a t o r s 13 defined on Q ~/2I-I which take Q ~/2H into E and a r e such that BQU2E2g2 (N, E). F o r B 6 ~ Q ( N , E) we set IBIQ = IIBQ1/2]I. It is known that if B6~2(H, E), then IBI -< [Igl!, B ~ Q (N, E) and[B [Q< [ B I(tr Q)~e. We r e t u r n to xe@l~or (R+, N). It can be shown that t h e r e e x i s t s a c o m p l e t e l y m e a s u r a b l e p r o c e s s Qx(t) with vatues in ~2 (N,'H) such that f o r a[[ (t, w) the o p e r a t o r Qx(t) is a s y m m e t r i c , n o n n e g a t i v e , n u e t e a r o p e r a t o r , while t r Q = 1 f o r a[l t, co, and
h~Q~ ff) h j
a (d x( x~,):d t )
(dPXd(x)t
a.s.)
f o r at[ i, j f o r any b a s i s {hi}, w h e r e dP x d (x}t is the differentia~ of the m e a s u r e
p. (A) = M ,f XA(t, c~) d ( x ) t, 0
defined on the p r o d u c t of ~" and the B o r e l a - a t g e b r a on [0, oo). We carl it the c o r r e t a t i o n o p e r a t o r of x. If B(t) is a c o m p [ e t e t y m e a s u r a b l e p r o c e s s in ff2(H,/~) and t
M~]iB(s)[12d(x ) , , ( o r
(a.s.) o f o r any t >- 0, then t h e r e e x i s t s a s q u a r e - i n t e g r a b i e m a r t i n g a l e y(t) in E which is (strongly) continuous in t such that f o r any o r t h o n o r m a l b a s e s {hi}and any y ~ E, T -> 0 l i m 3 i s u p gy(t)--
yB(s)h~d(h~x(s))
=0.
Two p r o c e s s e s y(t) p o s s e s s i n g this p r o p e r t y o b v i o u s t y coincide f o r all t (a. s.). to write
It is t h e r e f o r e c o r r e c t
t
(t) =j~ B (s) d x (s).
(2.1)
0
To c o m p u t e y(t) we fix b a s e s in H, E and set
~g(t)=Xe,g'(t),
g~(t)=~SeiB(s)h]d(h/x(s)) ,
t=l
j=l
0
w h e r e the c o n v e r g e n c e of the s e r i e s is u n d e r s t o o d as u n i f o r m m e a n - s q u a r e c o n v e r g e n c e in t on e a c h finite time i n t e r v a l [ i . e . , in L2(~, C([0, T], E)) f o r any T]. it is found that t
(v>t=~lB
(s) l2% ( J < x )
~,
(2,2)
0
and this t o g e t h e r with the ineqna[ity I B lQx -< IlBlq a f f o r d s the p o s s i b i t i t y of extending the s t o c h a s t i c i n t e g r M s (2.1) in the usual way with p r e s e r v a t i o n of the p r o p e r t y (2.2) to c o m p l e t e l y m e a s u r a b t e functions B(s) f o r which f o r att t ~_ o t
SllB(s)I?d < x ) s <
~,
o
The s t o c h a s t i c i n t e g r a t s a r e h e r e b y continuous in t and a r e [ocat m a r t i n g a l e s . cept of a s t o c h a s t i c i n t e g r a l to a still b r o a d e r c l a s s of p r o c e s s e s B(s).
It is p o s s i b l e to extend the c o n -
T H E O R E M 2.6. Suppose that f o r e a c h (s, co) t h e r e is defined an o p e r a t o r B(s)-~B(s, 0))Cff%(s.~0)(N, E) s u c h that B(s)Qlx/2(s) is a c o m p l e t e l y m e a s u r a b t e p r o c e s s [infg2(N, E)] and f o r e a c h t the r i g h t side of (2.2} is finite (a. s.). Then the s e q u e n c e
1241
t
[ 1 , ~II2(3))-ldx
(S)
0
c o n v e r g e s u n i f o r m l y with r e s p e c t to t in p r o b a b i l i t y to a l i m i t , s a y , y(t). m u l a (2.2) h o l d s .
Moreover,g(t)G~[or
E) and f o r -
F o r the p r o o f of t h i s t h e o r e m we n o t e t h a t Bn(S) - B(s)Q~/2{s)(1/n + Qlx/2(s))-~ is a c o m p l e t e l y m e a s u r a b l e p r o c e s s with v a l u e s in ~ (H, E),
HB=(s)[[4nlB(s)l%(s),
[B~(s)--B~(s)l~=
[B(s)Q~S2(s)e~(s)12[ 1 a,(s)_ _ ]-[ ~- + a i ( s ) ~+ai(s)
w h e r e e l ( s ) a r e the e i g e n v e c t o r s of Qx(s) and a~(s) a r e the c o r r e s p o n d i n g e i g e n v a l u e s . t h i s t h a t IBn(S) - B m ( s ) l Q -< ;B(S)IQ and IBn(s) - B m ( S ) l Q - - 0 a s n, m - - oo. H e n c e ,
It is e v i d e n t f r o m
t
< v~-v~ > ~=~ IB~ (s)--B~(s) l~(~)a < x > ~-~0, 0
w h e n c e the a s s e r t i o n s of the t h e o r e m a r e d e d u c e d in the w e l l - k n o w n w a y . S t o c h a s t i c i n t e g r a l s f o r f u n c t i o n s in s
(N, E) have p r e v i o u s l y b e e n d e f i n e d in [78].
The c o n s t r u c t i o n
p r e s e n t e d h e r e d i f f e r s s o m e w h a t f r o m the c o n s t r u c t i o n of the i n t e g r a l in [78]. The p r o c e s s y(t) in T h e o r e m 2.6 is t a k e n e q u a l to the r i g h t s i d e of (2.1) by d e f i n i t i o n . H i l b e r t s p a c e a n d ACZ (Z, X), then f o r at[ t (a. s.)
If X is a separable
t
Ag (0 =:S AB (s) dx (s).
(2.3)
0
We c h o o s e an e l e m e n t e ~ E and by m e a n s of it define an o p e r a t o r eC~ (E, R ~) b y the f o r m u l a ~y - e y (ey is the s c a l a r p r o d u c t in E). F r o m (2.3) we then have t
ev (t) = ~ ~B (s) d x (s). O
We o b s e r v e t h a t the o p e r a t o r 6B(s) on Qt/2H a c t s b y the f o r m u l a h ~ e B ( s ) h , while the l a t t e r is e q u a l to ( B ' e , h) if BE~(H, E). F i n a l l y , in the e a s e w h e r e h(s) ~ H, h(s) is c o m p l e t e l y m e a s u r a b l e , and f o r any t >- 0 t
t
[~ (~)I~(~)d ( ~ >~ = S I q~ ~ (~) h (~)t~d ( ~ ~s < ~ 0
(~. ~.),
0
we a g r e e to w r i t e t
t
I h (s) a x (~) = j' ~ (~) d x (s).
b~
0
We now i n t r o d u c e the c o n c e p t of a W i e n e r p r o c e s s in H. D e f i n i t i o n 2.8. L e t Q b e a n u c l e a r , s y m m e t r i c , n o n n e g a t i v c o p e r a t o r on H with t r Q < oo A W i e n e r p r o c e s s ( r e l a t i v e to {~-t}) in H with c o v a r i a n c e o p e r a t o r Q i s a c o n t i n u o u s m a r t i n g a l e w(t) with v a l u e s in H and c o r r e l a t i o n o p e r a t o r ( t r Q ) - l Q s u c h t h a t w(0) = 0, ( w } t = t r Q . t . It is known t h a t f o r any n u c l e a r , s y m m e t r i c , n o n n e g a t i v e Q with t r Q > 0 on a c e r t a i n p r o b a b i l i t y s p a c e it is p o s s i b l e to c o n s t r u c t a W i e n e r p r o c e s s c o r r e s p o n d i n g to it. It is c l e a r t h a t Mw2(t) = t r Q . t . S t o c h a s t i c i n t e g r a l s o v e r a W i e n e r p r o c e s s p o s s e s s e s p e c i a l l y good p r o p e r t i e s . F o r e x a m p l e , t h e y a r e d e f i n e d not o n l y f o r c o m p l e t e l y m e a s u r a b l e B(s) but a l s o f o r o p e r a t o r s m e a s u r a b l e in (s, co) w h i c h a r e : K ~ - c o n s i s t e n t and s u c h t
that
fiB(s)IQ2dS~ ~ ( a . s . ) f o r any t --> 0. ,o
We c o n c l u d e t h i s s e c t i o n w i t h the r e m a r k t h a t in p l a c e of an infinite t i m e i n t e r v a l a b o v e we could c o n s i d e r a s e g m e n t of the f o r m [0, T]. In o r d e r to f o r m a l l y have the p o s s i b i l i t y of d o i n g t h i s , it s u f f i c e s to e x t e n d the p r o c e s s e s in q u e s t i o n to t _> T b y s e t t i n g t h e m e q u a l to the v a l u e w h i c h t h e y a s s u m e at t = T. 3.
Ito
Formula
for
the
Square
of
the
Norm
L e t V be a B a n a c h s p a c e , let V* b e the d u a l s p a c e of V, and l e t H b e a H i l b e r t s p a c e (we a s s u m e that t h e y a r e r e a l s p a c e s ) . If v E V (h E H, v* e V*), t h e n ivi ( i h l , Iv*i) d e n o t e s the n o r m of v(h, v*) in V(H, V*); 1242
if h 1, h 2 6 H, then hlh 2 d e n o t e s the s c a l a r p r o d u c t of h i , h2; the r e s u l t of the a c t i o n of a f u n c t i o n a l v* ~ V* on an e l e m e n t v E V we w r i t e vv* = v*v. L e t A b e a b o u n d e d , l i n e a r o p e r a t o r a c t i n g f r o m V to H s u c h t h a t AV i s d e n s e in H. We c o n s i d e r t h r e e p r o c e s s e s v(t, o~)CV, h(t, o~)@I, v*(t, oJ)~V* d e f i n e d f o r t -> 0 on s o m e c o m p l e t e p r o b a b i t i t y s p a c e (f~, ~r: p) and c o n n e c t e d w i t h s o m e e x p a n d i n g f a m i l y of c o m p l e t e a - a l g e b r a s ~ r - t ~ , - t.~O i n a p a r t i c u l a r w a y . L e t v(t, co) be s t r o n g l y m e a s u r a b l e (in the L e b e s g u e s e n s e ) with r e s p e c t to (t, co) and w e a k l y m e a s u r a b l e with r e s p e c t to co r e l a t i v e to f t f o r a l m o s t e v e r y t; f o r any v ~ V the q u a n t i t y vv*(t, co) i s .~t. m e a s u r a b l e in co f o r a l m o s t e v e r y t and is m e a s u r a b l e in (t, co). It is a s s u m e d t h a t h(t, co) is ( s t r o n g l y ) c o n t i n u o u s in t , is s t r o n g l y m e a s u r a b l e in co r e l a t i v e to Art f o r e a c h t , and i s a l o c a l m a r t i n g a l e . The t a t t e r m e a n s t h a t in H t h e r e e x i s t s t r o n g l y A r t - m e a s u r a b l e p r o c e s s e s A ( t ) , re(t) w h i c h a r e c o n t i n u o u s in t s u c h t h a t re(t) is a l o c a l m a r t i n g M e , the t r a j e c t o r i e s Aft, c~) (for e a c h co) have finite v a r i a t i o n on b o u n d e d t i m e i n t e r v a l s , a n d h(t) = Aft) + m ( t ) . We fix p ~ (1, o~) and s e t q = p / ( p - 1). We a s s u m e t h a t I v ( t ) I ~ p ( [ 0 , r ] ) ( a . s . ) f o r any T -> 0 and t h e r e e x i s t s a function f(t, w) m e a s u r a b l e in (t, co) s u c h t h a t f(t) 6~q~ T]) (a.s.) f o r any T > 0, and Iv*(t)l -< f(t) f o r a l l (t, co). R e g a r d i n g the l a s t c o n d i t i o n it is u s e f u l to n o t e t h a t I v*(t)l i s , g e n e r a t i y s p e a k i n g , not m e a s u r a b l e . T h i s n o r m is m e a s u r a b l e , e . g . , if V i s s e p a r a b l e , and in t h i s c a s e I v* (t) I ~:Fq([0, r ] ) (a. s.) f o r any T_>0. We f o r m u l a t e the m a i n r e s u l t r e g a r d i n g I t o ' s f o r m u l a . T H E O R E M 3.1. t <
L e t r b e a M a r k e r t i m e and s u p p o s e t h a t f o r e v e r y v ~ V a l m o s t e v e r y w h e r e on {(t, co) :
~(~1} f
AvAv (0 = ! vv* (s) ds + Avtz (t). 'o Then t h e r e e x i s t a s e t 9 ~ ' ~
(3.!)
and a function h(t) w i t h v a l u e s in H s u c h t h a t
a) P(fZ') = 1, h(t) is s t r o n g l y 5 ~ - m e a s u r a b t e on the s e t {co : t < T(o))} f o r any t , h(t) is c o n t i n u o u s in t on [0, ?(co)) f o r e v e r y c~, and Av(t) - h(t) [ a . s . {(t, c o ) : t < r(co)}l; b) f o r co r C/', t < r(w) fz~ ( 0 =-
h~ (o) + 2 ,I v (s) ~* (s) ds + 2 f ~ (x) d/z (s) + < .~ > ~; 0
(3.2)
\0
c) i f V is s e p a r a b t e , then f o r c o 6 ~2', t < T(co), v ~ V Avh(t)=
I vv* (s) ds@Avh(t);
(3.3)
d) if V is s e p a r a b l e and (3.1) is s a t i s f i e d f o r s o m e t >- 0 and e a c h v E V ( a . s . ) on {w : t < T(co)}, then Av(t) = h(t) (a. s.) on {co : t < T(co)}. We t a k e up the p r o o f of t h i s t h e o r e m a f t e r d i s c u s s i n g i t s h y p o t h e s e s and a s s e r t i o n s . F o r the e x i s t e n c e of the s t o c h a s t i c i n t e g r a l in (3.2) it s u f f i c e s t h a t h(s) b e c o m p l e t e l y m e a s u r a b l e a n d f o r t < T(CO)
f l~(s)ld is c o n t i n u o u s in s and is Y % - c o n s i s t e n t , while <m}t + LAItt < oo. We p o i n t out t h a t b y a s t o c h a s t i c i n t e g r a l we s h a l l a l w a y s u n d e r s t a n d a c o n t i n u o u s (for a l l w) p r o c e s s . F u r t h e r , s i n c e v(s) is s t r o n g l y m e a s u r a b l e , w h i l e v*(s) is w e a M y m e a s u r a b l e , v(s)v*(s) is m e a s u r a b l e in (s, co), and b y o u r a s s u m p t i o n s it is l o c a l l y t n t e g r a b l e in s (a. s . ) . A l l e x p r e s s i o n s in (3.2) a r e t h e r e f o r e meaningful. A s s e r t i o n d) is a s i m p l e c o r o l l a r y of the p r e c e d i n g a s s e r t i o n s . I n d e e d , f r o m (3.1) and (3.3) we h a v e Avh(t) = AvAv(t) (a. s.) on {co : t < r(co)} f o r any v e V. But V i s s e p a r a b l e , and t h e r e f o r e Avh(t) = AvAv(t) f o r
1243
a l l v EV ( a . s . ) on {w : t < r(w)}.
S i n c e AV i s d e n s e in H, we i m m e d i a t e l y o b t a i n f r o m t h i s the a s s e r t i o n d).
We now s u p p o s e t h a t we have p r o v e d the t h e o r e m in the e a s e w h e r e H and V a r e s e p a r a b l e . We s h a l l show t h a t it is then p o s s i b l e to o b t a i n i t s a s s e r t i o n s in the g e n e r a l c a s e . Since p r o c e s s v(t) is s t r o n g l y m e a s u r a b l e in (t, co), t h e r e e x i s t s a p r o c e s s vT(t) w h i c h c o i n c i d e s with v(t) f o r a l m o s t a l l (t, w) and h a s i t s r a n g e in s o m e s e p a r a b l e s u b s p a e e V ' c V . It i s shown s i m i l a r l y that on a s e t of full p r o b a b i l i t y 12" at[ the v a l u e s of h(t, co), t >- 0, w E I2" lie in s o m e s e p a r a b l e s u b s p a e e H ' c H . We m a y a s s u m e with no l o s s of g e n e r a l i t y t h a t I2" = ~ . By h y p o t h e s i s , AV i s d e n s e in H. H e n c e , t h e r e e x i s t s a s e p a r a b l e s u b s p a e e V " c V sueh t h a t H ' ~ A V " . S u p p o s e now t h a t V, is the c l o s e d s p a c e s p a n n e d b y V'UV" and H1 is the c l o s e d s p a c e s p a n n e d b y AVI. The s p a c e s V1, H1 a r e s e p a r a b l e , and AV1 is d e n s e in H~. F u r t h e r , v ' ( t ) ~ V ~ , h(t){H~,while the f u n e t i o n a l s v*(t) on V a r e a l s o f u n e t i o n a l s on V1. It m a y t h e r e f o r e b e a s s u m e d t h a t v*(t) E V~. The n o r m of v*(t) in V~ is h e r e b y no l a r g e r than I v * { t ) l v , . R e l a t i o n (3.1) is p r e s e r v e d f o r any v E V~ (even f o r v E V), s i n c e v(t) = v'(t) [ a . s . (t, co)]. H e n c e , if T h e o r e m 3.1 is t r u e f o r s e p a r a b l e V and H, then we o b t a i n its a s s e r t i o n in the g e n e r a l e a s e b y a p p l y i n g it to V1, H 1. We m a y thus a s s u m e with no t o s s of g e n e r a l i t y until the end of the s e c t i o n t h a t V and H a r e s e p a r a b l e . We s h a l l e x p l a i n why (3.2) i s c a l l e d I t o ' s f o r m u l a f o r the s q u a r e of the n o r m . a l l p r o c e s s e s v(t), h(t), v*(t) in a s i n g l e s p a c e .
F o r t h i s p u r p o s e we p l a c e
In t h o s e e a s e s w h e r e the s a m e v e c t o r b e l o n g s to v a r i o u s s p a c e s we equip its n o r m with the s y m b o l of the s p a c e in which it is c o n s i d e r e d . S u p p o s e t h a t the s p a c e V is a ( p o s s i b l y , n o n e l o s e d ) s u b s p a e e of H w h i c h is d e n s e in H (in the n o r m of H) a n d I vlH -< N I v I v f o r a l l v E V, w h e r e N d o e s not d e p e n d on v. S u p p o s e t h a t H i s , in t u r n , a s u b s p a c e of s o m e B a n a c h s p a c e V ' and t h a t H i s d e n s e in V ' . Then
Vcttc
V'.
(3.4)
We a s s u m e t h a t the s c a l a r p r o d u c t in H p o s s e s s e s the f o l l o w i n g p r o p e r t y : if ~p E V, ~ E H, then I ~o~bI 5 I~IVol ~lV T. S i n c e the i m b e d d i n g s in (3.4) a r e d e n s e , it is p o s s i b l e to u n i q u e l y d e f i n e g0{bfor~o E V, ~b ~ V ' a s l i m ~ n w h e r e ~n E H, I~ - ~nlV' ~ 0. O b v i o u s l y , f o r ~ E V ' the e x p r e s s i o n g0~ i s a b o u n d e d l i n e a r f u n e tz-+~
t i o n a l on V. We s u p p o s e t h a t f o r ~b E V ~ the e q u a l i t y go~b = 0 f o r a l l ~0 E V i m p l i e s t h a t ~ = 0. Then the m a p p i n g which a s s i g n s to e l e m e n t s ~ E V Tthe f u n c t i o n s qg~bon V is a o n e - t o - o n e m a p p i n g of V into s o m e s u b s e t of V* We a s s u m e t h a t any b o u n d e d , l i n e a r f u n c t i o n a l on V h a s the f o r m ~o$, w h e r e ~ E V T. Then the m a p p i n g V ' ~ V* m e n t i o n e d a b o v e is a o n e - t o - o n e m a p p i n g of V ' onto V* and V ' can be i d e n t i f i e d with V* if d e s i r e d . We note t h a t u n d e r t h i s i d e n t i f i c a t i o n the n o r m s of ~ a s an e l e m e n t of V ' and a s an e l e m e n t of V* a r e g e n e r a l l y d i s t i n c t , and we s h a l l n o t i d e n t i y V ' with V*.~ F i n a l l y , we s u p p o s e t h a t a function v ' ( t , co) is d e f i n e d with v a l u e s in V ' such t h a t vvT(t) is ~r'e - m e a s u r a b l e in co, is L e b e s g u e - m e a s u r a b l e r e l a t i v e to (t, co) f o r any v ~ V, and I v ' ( t ) l v , -< f(t) f o r a l l (t, w). For
any vEV,
t-> 0
i
vv'
(s) ds ...<1v Iv ~t f (s) ds. 0
From
the properties
of f(t) it therefore
follows
that there
exists a set
!2'~
such
that P(f2') = 1 and for
t
w ~ f~', t -> 0 the i n t e g r a l S~~
ds is a b o u n d e d , l i n e a r f u n c t i o n a l on V.
U n d e r o u r a s s u m p t i o n s it can be
O
w r i t t e n in the f o r m v~b(t), w h e r e ~(t) E V ' .
F o r co ~ f~, we s e t by d e f i n i t i o n l
tp ( t ) = ~ v' (s) ds. o T h e o r e m 3.1 now a c q u i r e s the f o l l o w i n g f o r m . THEOREM
3.2.
For
w E ~2', t-> 0we
set
(3.5)
t
(t) = I v' (s) ds + h (t) 0
t This means tion.
1244
in the present
section.
On the contrary,
in subsequent
chapters
we
shall make
this identifica-
and we s u p p o s e that h(t) : v(t) for a l m o s t art (t, co). Then t h e r e e x i s t s a s e t ' ~ " c J " s u c h that P(~]';) - 1 and for co ~ ~ " the f u n c t i o n t](t) t a k e s v a l u e s in I~I, is c o n t i n u o u s in H with r e s p e c t t o t , i s ( s t r o n g l y ) ~ ' t - m e a s u r a b t e with r e s p e c t to co for e a c h t (as a f u n c t i o n with v a l u e s in H), and
Yz2(t):IF(O)@2J v(s)v'(s)ds+2j [z(s)dh(s)@ < m ) t. 0
0
F o r s i m p l i c i t y of the f o r m u l a t i o n , we have t a k e n r = ~o. a r b i t r a r y r is o b v i o u s .
The p o s s i b i l i t y of g e n e r a l i z i n g to the c a s e of
We note that r e l a t i o n (3.6) is o b t a i n e d if the r u l e s f o r c o m p u t i n g the s t o c h a s t i c d i f f e r e n t i a [ dh2(t) for the p r o c e s s h(t) d e f i n e d in (3.5) a r e a p p l i e d . One of the d i f f i c u l t i e s in j u s t i f y i n g (3.6) is that it is g e n e r a l l y not c l e a r why h(t) ~ H [why h 2 ( t ) e x i s t s ] , s i n c e Eq. (3.5) o n l y d e f i n e s h(t) as a p r o c e s s w i t h v a l u e s in V', and it is g e n e r a l l y not t r u e that h(t) ~ V f o r all (t, co). P r o o f of T h e o r e m 3.2. We take for A the i d e n t i t y o p e r a t o r and use the fact that the f o r m u l a t i o n of T h e o r e m 3.2 does not c o n t a i n the p r o c e s s v*(t). We c o n s t r u c t a p r o c e s s v*(t) on the b a s i s of v'(t) as the p r o t e s s c o r r e s p o n d i n g to v'(t) u n d e r the m a p p i n g V ' - - V*. Then v*(t) s a t i s f i e s the a s s u m p t i o n s p r e c e d i n g T h e o r e m 3.1, and (3.1) follows f r o m (3.5). On the b a s i s of T h e o r e m 3.1, we c o n s t r u c t a set ~q"c2' and a p r o c e s s hi(t) s u c h that P ( ~ " ) - 1, [~(t) is c o n t i n u o u s in H with r e s p e c t to t, is an S t - m e a s u r a b l e p r o c e s s with r e s p e c t to co, and hi(t) - v(t) f o r a h n o s t all (t, co); for co e ~" f o r m u l a (3.6) holds if h is r e p l a c e d by h;, and for co ~ s v ~ V , t->..0 t
~ ' (t) = , [ z,~, (s) ~/s + ~/, (t) = zT,~(t). 0
F r o m the e q u a l i t y of the e x t r e m e t e r m s , it follows that v(hl(t) - h(t)) = 0 f o r all v e V, and hence h~ : for t -> 0, co e ~2"; f r o m the p r o p e r t i e s of ~1 we now o b t a i n all the r e q u i r e d p r o p e r t i e s of h. The p r o o f of the t h e o r e m is c o m p l e t e . We p r o c e e d to p r e p a r e for the proof of T h e o r e m 3.1. We have a l r e a d y a g r e e d to c o n s i d e r V and hence also H to be s e p a r a b l e . F u r t h e r , if in place of r in T h e o r e m 3.1 we take ~A a, p r o v e the t h e o r e m for * A n, a n d then let n - - 0o, then we o b t a i n the proof of the t h e o r e m for r . It m a y t h e r e f o r e be a s s u m e d that r is a b o u n d e d M a r k o v t i m e , and if a n o n r a n d o m change of t i m e is a p p l i e d e v e r y t h i n g r e d u c e s to the e a s e r -< 1. We f u r t h e r note that Iv(s)I, Iv* (s)l a r e Y ' ~ - m e a s u r a b l e for a l m o s t all s and a r e L e b e s g u e - m e a s u r a b l e in (s, co). H e n c e , the p r o c e s s f
r (t):h2 (0)@I[ n lit+ < m ) t+5]'o
t
(s)lP ds+Sl~* (s)lq as
0
l b r e a c h t is Y - r - m e a s u r a b l e and is c o n t i n u o u s in t (a. s 3.
0
T h i s i m p l i e s that for any n
, ( n ) = ini { t > 0 : r ( t ) > n } h
z
is a M a r k o v t h n e . Since r(n) ~ r , it s u f f i c e s to p r o v e T h e o r e m 3.1 with r in its f o r m u l a t i o n r e p l a c e d by r ( n ) . M o r e o v e r , p r o c e s s r(t) is b o u n d e d in (t, co) on {(t, co) : t ___ r ( n , co)}, and it m a y t h e r e f o r e be a s s u m e d in the p r o o f of T h e o r e m 3.1 that the p r o c e s s r(t) is b o u n d e d on {(t, co):t -< r(co)}. S e t t i n g , if n e c e s s a r y , v(t) = 0, v*(t) = 0, h(t) = h(r) for t _> r, we a r r a n g e that p r o c e s s r(t) is b o u n d e d on [0, <) x ~. A f t e r t h i s , on m u l t i p l y i n g (3.1) by a s u i t a b l e c o n s t a n t e and r e p l a c i n g v, v*, h by gv, sv*, eh, we r e d u e e the m a t t e r t o t h e c a s e w h e r e r g ) - 1.
T h e s e a r g u m e n t s show that it s u f f i c e s to p r o v e T h e o r e m 3.1 in the s p e c i a l e a s e c o n s i d e r e d in the n e x t section.
4.
Proof
of T h e o r e m
3.1
In this s e c t i o n we p r o v e T h e o r e m 3.1 u n d e r the a d d i t i o n a l a s s u m p t i o n s that V and H a r e s e p a r a b l e , and
for all co, r -< 1, 1
1
[~ (s)[rcls+f [~* (s)'j ds§ A [[~i 0
< .z > 1@-~ 2 (0) ~.~ 1 .
(4.1)
0
It is shown in the p r e c e d i n g s e c t i o n that t h e s e a s s u m p t i o n s i n v o l v e no l o s s of g e n e r a l i t y . LEMMA 4.1. T h e r e e x i s t s a s e q u e n c e of i m b e d d e d p a r t i t i o n s 0 - t~ < t~ < . .. < t[(n)+ 1 = ! of the s e g m e n t [0, 1] with d i a m e t e r t e n d i n g to zero and t h e r e e x i s t s a set ~2'c2 p o s s e s s i n g the following p r o p e r t i e s :
1245
1) P ( ~ ' ) = 1 f o r w e f t ' , t < T ( w ) , t e I {tn;i= v ~ V, f o r u ~ I the q u a n t i t y v(u) is 5 , - m e a s u r a b l e ;
1.....
k(n), n - > 1}, e q u a l i t y (3.1) is s a t i s f i e d f o r a l k
2) we s e t v n(1)(0 v(t n) f o r t 6 [tl~, tn+l), i = 1, . . . . k(n), Vn0)(t) = 0 f o r t 6 [0, t~), V(n2)(t) = v(tn+~) f o r t [tn , ti+t), n n i = 0 . . . . . k ( n ) - 1, v n( 2)(t) = 0 f o r t e [tk(n) , 1); then f o r j = 1 , 2 1
qd(nJ)(t)]P
M sup t~[O,1
n~
(4.2)
0
P r o o f . L e t [a] be the i n t e g r a l p a r t of the n u m b e r a, and let >d(n, t) = 2 - n [ 2 n t ] , ~2(n, t) = 2-n[2nt~ + 2 - n , v(t) = 0 f o r t ~ [0, 1]. The u s e of s t a n d a r d a r g u m e n t s of Doob s h o w s t h a t t h e r e e x i s t s a s e q u e n c e of i n t e g e r s r n ~ oo s u c h t h a t f o r a l m o s t a l l s E [0, 1], j 1, 2 I
lira M ~ [ v (t) - - v (xJ (r~, t + s) -- s) ]P d t = 0. rtz~ 0~
(4.3)
F u r t h e r , it f o l l o w s f r o m the s e p a r a b i l i t y of V and F u b i n i ' s t h e o r e m t h a t t h e r e e x i s t s a s e t T ~ [ 0 , 1] of unit L e b e s g u e m e a s u r e s u c h t h a t f o r t E T and a l l v E V (3.1) i s s a t i s f i e d (a. s.) on { w : t < T(W)}, and the q u a n t i t y v(t) i s ~ ' i - m e a s u r a b l e . It is c l e a r t h a t f o r any s 6 [0, 1] a l l v a l u e s of the f u n c t i o n s KJ(rn, t + s) - s f o r t 6 [0, 1], j = 1, 2, n -> 1, l y i n g in [0, 1] a l s o b e l o n g to T. We fix a s u i t a b l e s so t h a t (4.3) is a l s o s a t i s f i e d ; we define {tn} a s the s e t of v a l u e s of ~ l ( r n , t + s) - s f o r t 6 [0, 1] which tie in [0, 1 ] , t o w h i e h we a d d the p o i n t s 0 and 1, and we d e n o t e b y ~2~ the s e t of co f o r w h i c h Eq. (3.1) is s a t i s f i e d f o r a l l v 6 V, t = t ni < z(w), i = 1, . . . . k(n), n -> 1. A l l a s s e r t i o n s of the l e m m a a r e then v a l i d e x c e p t p o s s i b l y f o r the f i r s t i n e q u a l i t y in (4.2). We n o t e , h o w e v e r , t h a t b y v i r t u e of the s e c o n d i n e q u a l i t y in (4.2) f o r s u f f i c i e n t l y l a r g e n 1
.44 f I v~J) (t)
l" at < oo.
0
This inequality is equivalent to the first inequality in (4.2), whichis thus valid for large n. clearty valid since our partitions are imbedded. The proof of the lemma is complete. L E M M A 4.2.
For w~f2',
For small n it is
t , s 6 I , s - < t < T(W)
I Av (t) I~ - I Av (s) ? = 2 f v (t) v* (u) du + 2Av (s) (k (t) 8
- h (s)) § ] h ( t ) - h (s) I~ - IA (v ( t ) - v (s)) - (~ ( t ) - h (s)) ?,
(4.4)
t
(t) 1~=21 v (t) v* (u) d ~ §
(t)--[ Av (0--Zz (t)?.
(4.5)
(;
The p r o o f of t h i s [ e m m a is b a s e d on u s i n g (3.1) f o r v = v(t), v(s) and s i m p l e a l g e b r a i c t r a n s f o r m a t i o n s w h i c h we l e a v e to the r e a d e r w h i l e s u g g e s t i n g that (4.5) be d e r i v e d f i r s t and (4.4) then p r o v e d b y s u b t r a c t i n g the a p p r o p r i a t e e q u a l i t i e s (4.5). L E M M A 4.3.
:v/ sup Proof.
i Av (t)[: < o0.
F r o m (4.5), (4.2) f o r t e I t
Mz,<~l a~(t)i~-,<M~ 2 (t)+~tMl~(t)i, ~ M f [~*(u)lqa~< o~.
(4.6)
0
F u r t h e r , b y (4.4), (4.5) and t = t ni < ~(co), w e Q~ t
t
lag ') (t)? =h ~ Co) + 21 G2) (") v* (u) d. +2 f AG1) (u) dh (u) +2h (0) (tz(tg) --h (0)) 0
i--1
0
i--1
§ Z Ih (t]+i) - - h (t~)[2_ ~ I A (v (t]+~) -- v (t~)) -- (h (t]+~) - - h (t]))]2, j'=0
]=0
w h e r e in the l a s t s u m the t e r m c o r r e s p o n d i n g to j = 0 is t a k e n e q u a l to I Av(t n) - h(t n) 12. To e s t i m a t e the s e c o n d t e r m on the r i g h t s i d e of (4.7), we a p p l y (4.1) and the B u r k h o l d e r i n e q u a l i t y . We then o b t a i n
1246
(4.7)
n
i
s.p x,,,<, 0t~l A<') (") ah (It) ~<~ s pl,lt i i)t
(.)
i
..< M
,>~supI Av (tT) l x,7<, II k ]/, + aM g.<,}Av(~~) (u)t2d
(.)l
<m } = 2
/~n'l 12
~< 4M/>,sup 1k v (t ?)I 9~,~<,~< 1 M sup A m l, > , t J t Z ,7<, v- 16. We note t h a t it f o l l o w s f r o m (4.6) t h a t the l a s t e x p r e s s i o n is f i n i t e .
Moreover,
f r o m (4.1)
k(n)
M Z I h (ts+~) -- a (Is) i" -% 4td II a Ii; + 2M't~ (1) < 6. ]=0
F r o m t h i s and f r o m (4.7) I
0
L e t t i n g h e r e n ~ ~o and u s i n g the f a c t t h a t b y (4.2) the r i g h t s i d e is b o u n d e d w i t h r e s p e c t to n , we o b t a i n t h e a s s e r t i o n of the [ e m m a . The p r o o f of the [ e m m a i s c o m p l e t e . L E M M A 4.4.
We s e t
Then f o r co e 12" on [0, r ] t h e r e is d e f i n e d a function h(t) w i t h v a l u e s in H w h i c h is w e a k l y c o n t i n u o u s in H with r e s p e c t to t and s a t i s f i e s (3.3) f o r a l l co ~ f~", t < r(co), v e V . M o r e o v e r , h(t) is Y t - m e a s u r a b t e on {co : t <
r(co)}, ~(t) = av(t) ( a . s . ) o n { ( t , co):t < r(co)}, ~(t)
av(t) f o r c o ~ a " ,
tel
t
P r o o f . F o r co ~ ~ " , t 6 I, t < r , v ~ V the function AvAv(t) c o i n c i d e s with the r i g h t s i d e of (3.3) w h i c h is c o n t i n u o u s in t. H e n c e , f o r s -< r t h e r e e x i s t s lira AvAv(t).
l~t~s
S i n c e [ Av(t)l is b o u n d e d on I f~ {~ < ~}, t h i s m e a n s t h a t f o r s ..
h(r) f o r t - >
r, h(t)- 0forc0r
and we note t h a t f o r c o e f ~ "
(4.s)
sup ifi(t) l-- sup IA~ d) l< oo.
t Moreover,
f o r t -< 1 t h e r e e x i s t s f ~(s) dh (s).
In o r d e r to p r o v e t h i s , it s u f f i c e s to e s t a b l i s h t h a t ,h(s) is
0
completely measurable. T h i s p r o p e r t y of h(s) f o l l o w s f r o m the s e p a r a b i l i t y of H and the c o n t i n u i t y of 7rh(s) f o r co 6 ~2", w h e r e % i s the o p e r a t o r of p r o j e c t i n g onto any f i n i t e - d i m e n s i o n a l s u b s p a e e of H. L E M M A 4.5.
We set hn(t) = ~(t n) for t ~ [~, t ni + l ) , i = 0 . . . . . sup s
s
t
k(n).
~ a~ (s) dh ( s ) - l fz (s) dh (s) 00
Then a s n ~ oo we have in p r o b a b i l i t y -+-0,
(4.9)
0
P r o o f . L e t h~ . . . . , h r , . . . b e an o r t h o n o r m a l b a s i s in H, and l e t rrr b e the o p e r a t o r p r o j e c t i n g H onto the s p a c e g e n e r a t e d b y h, . . . . . h r . Since ~rl~(s) i s c o n t i n u o u s on f~", it f o l l o w s t h a t (a. s.) lira
Irtrhn(s)--xrh(s)12d ( t,t > t -}-j ]arfi~(s)--.%[i(s)[diiA[l,
=0.
It t h e r e f o r e s u f f i c e s to p r o v e that f o r any e > 0 litn sup P sup t"' (1 - - a,) h~ (s)
a~Iz(s) > 2~} =
0,
(4.10)
1247
O,
lim P r-+~
{ t
0
We shall prove only the f i r s t equality. The s e e o n d is p r o v e d s i m i l a r l y . o p e r a t o r , f o r any N, 6 > 0 we e s t i m a t e the p r o b a b i l i t y in (4.10) in t e r m s of
P/sup ( t,~i
/r
i}
ltn(s)a(l--a,r)k(s ) >2~ < ~ + P
o
< ~ 472P {sup ]/7(s) ] >
(sfh.(s)ldll(1--~*~)Atl,>~ 47p
Noting that rrr is a s e i f - a d j o i n t
}
]h.(s)12d((1--a*c)m>,>5 ...<
N} 47 N 21411(1 --nr) A tJl 47 TN N I(1 - a t ) m (1) 12.
(4.11)
s~I
We denote by h i the i-th c o o r d i n a t e of h e H in the b a s i s {hi}, and we set ai(t) = dAi(t)/d IIA Ht. r
Then as
~cr 1
I
j(1 --~,) m ( 1 ) i 2 = ~
(m' (1))2-+ 0.
i>r
F r o m this and f r o m (4.11) and (4.1) we see that the left side of (4.10) does not e x c e e d ~ 472P {sup l/7(s) I > N } . g
s~l
Since 6, N a r e a r b i t r a r y and (4.8) is s a t i s f i e d , the last e x p r e s s i o n can be m a d e a r b i t r a r i l y s m a l l . p r o o f of the [ e m m a is c o m p [ e t e .
The
We now define the set ~2"T which will play the rote of f2 T in T h e o r e m 3.1. It is possible to find a sequence {n'} along which the left side of (4.9) tends to z e r o (a. s.). It m a y be a s s u m e d with no loss of g e n e r a l i t y that the o r i g i n a l s e q u e n c e has this p r o p e r t y . M o r e o v e r , we set % = 0; then, as is well known, in p r o b a b i l i t y f o r r -> 0, t ~ [0, iI lira ~ t
I(l--~,)(h(t;+~)--h(tgl 2= < ( l - - r ~ r ) m } t .
0.12)
j+l~.t
T h e r e f o r e , t h e r e e x i s t s a s u b s e q u e n c e along which the last equality u n d e r s t o o d in the sense of pointwise c o n v e r g e n c e is t r u e f o r air r -> 0, t ~ [ a l m o s t s u r e l y . To s i m p l i f y the notation, we a s s u m e that this s u b s e quenee also c o i n c i d e s with the o r i g i n a [ s e q u e n c e . We set
Q 2 = ~ ~ fl/m:lim
= till / t
~l(1--~0(h
(tj+~)-j ) ) [2 = ( ( 1 - - a , ) m ) ~ h (t"
t},
n~ootn
"an=~ z=1,2,
Qa=/re:lira S I%~
Q4= {o~:lim < 1 - - ~ ) m ) 1=0}, r~ 4
e ' = 2" n NQ, and define h(t) off ~ " by setting h(t, w) = 0 f o r w r ~'". F r o m what has b e e n said a b o v e , P(Qi) = 1, i 1, 2. F r o m L e m m a 4.1 it follows that it m a y be a s s u m e d that P(Q3) = 1. Since the s e q u e n c e {(1 - 7rr)m}l is d e c r e a s i n g (a.s.) and M ( (1 -- ~r) "~ ) 1~ .3d I (1 -- at) rrL(1) 12-+ 0, it follows that/) (Q4) = 1. T h u s , P(S2'") = 1.
1248
F o r w ~ ~2"', t , s
LEMMA 4.6.
~I,
s < t < Tiw)
t
t
i ~ (t) -- fi i s) I~ = 2 f (v (tt) -- v (s)) v* (u) dtt @ 2 i ('~ (u) -- h (s)) dh (tt) @ ( .~ ) t -- ( m ) ,,
(4.12)
s
s
/ ~ (01 ~ = t: io) + 2oS v (u) v* (~) du + 2
i ~ (.) d/z (u~)+ ( ~ ) ,.
(4.13)
0
P r o o f . We f i r s t p r o v e (4.13). We fix co ~ f2'", t ~ t, t < r(co). n n t is a p o i n t of t h e p a r t i t i o n t o < . . . < tk(n)+ D we h a v e t
tt ~ ( t ) = h : ( 0 ) + 2 S ~I;~) (u) v* ( . ) d t t + 2 0
i h~(u) dh (u)-t 0
,%:,
From
(4.7) a n d L e m m a
4.4 f o r n s u c h t h a t
Ih (t~+~)-- h (ty)12_ ~ [(~ (t]+~)-- h (ty)) ,':+:,
- i,'* it%,) - h (t~))5 Letting
n ~
00, w e find t
it)
h 2 i0) +
t
--I,
2 o u
0
where
I = lira E
](~ (ty+,) --/z (fj))
- -
(h (tj%,)
- - h (tT))12,
/z~oo
t]+t~t
a n d the l a s t l i m i t e x i s t s a n d is f i n i t e . F o r the p r o o f of (4.13) it s u f f i c e s to s h o w t h a t I = 0. It a s s t u m e t h a t the b a s i s { h i } is f o r m e d f r o m e l e m e n t s of t h e s e t AV. S i n c e the l a t t e r is d e n s e in a s s u m p t i o n i n v o l v e s no l o s s o f g e n e r a l i t y . M o r e o v e r , it is o b v i o u s t h a t f o r a n y r >- 1 t h e r e is c o n t i n u o u s in t h e n o r m of V s u c h t h a t A r t ( t ) ~rrh(t) f o r a l l (t, w). On t h e b a s i s of t h e f u n c t i o n f u n c t i o n s [ ( i ) n , i = 1, 2 j u s t a s in L e m m a 4.1 t h e f u n c t i o n s g~i) a r e c o n s t r u c t e d on t h e b a s i s of b y ( 3 . t ) and L e m m a 4.4 f o r tj+~ n ~ t a n d any (o E V t h e r e is the e q u a b l y
i s c o n v e n i e n t to H, t h i s l a s t a function ~r(t) Vr we c o n s t r u c t v. N o t i n g t h a t
*z
t]+t
((h ()%3
-/7
(tT)) - ih (tM)
-/~
iC)))A~ =
f wv* (u) du, l Ii ]
we find easity for any r >- 0 t
!
n-+~
:I~
((~iC+3-~(t])) t r:, .42 t
0
Jtt
- (h i%,) - h (t;))) il - ~r) (h it M ) - h itg). H e r e the f i r s t l i m i t is e q u a l to z e r o ; s i n c e co ~ Q3, the s e c o n d is e q u a l to z e r o Vr(l!n(u)I ~ 0 uniformlv~ with respect to u due to the continuity of ~r(U).
because '
I~(~) r,n (u)
-
Hence, l
1
I
A s r ~ ~ t h i s i m p l i e s t h a t J = 0. E q u a l i t y (4.13) h a s b e e n p r o v e d . (4.13) b y m e a n s of the r e l a t i o n s (a - b) 2 = a 2 - b ~ - 2b(a - b), t
- 2 h is) (~ i t ) - h is))= - 2 ,f v (s) v* (u) a . - 2 3
T h e p r o o f of the l e m m a
E q u a l i t y (4.12) is d e d u c e d f r o m t
~ ? is) dh (.). 3
is c o m p l e t e .
1249
We now f i n i s h the p r o o f of T h e o r e m 3.1 in the s p e c i a l c a s e u n d e r c o n s i d e r a t i o n . B e c a u s e of L e m m a 4.4 and (4.13) it r e m a i n s f o r us to p r o v e the s t r o n g c o n t i n u i t y of h(t, co) in t f o r t < T(CO), co e t2"'. Since a w e a k l y c o n t i n u o u s function with a c o n t i n u o u s n o r m is s t r o n g l y c o n t i n u o u s , it s u f f i c e s to p r o v e (4.13) f o r t < T(co), co e l 2 " . F o r t = 0 (4.13) is o b v i o u s . We f i x t > 0, t < r(co), co el2"'. F o r a l l s u f f i c i e n t l y l a r g e n it is p o s s i b l e to define j = j(n) s u c h t h a t 0 < t j -< t < t j + l . and note that t(n) r t , t
We s e t t(n) = tj(n)
1
lim ff i v (u) -- v (t (n))]. l v* (u)] du ..< lira ~ I v (u) -- v}, ~) (u) l 9[ v* (u)l du -O, n~
t(n)
l im sup
n~
h (u) - - h (t (n)) dh
0
-,< 2 litn sup
(u))
lira ( ( m > t-- < rn > ,~,,)) =0. Therefore,
t h e r e e x i s t s a s u b s e q u e n e e n(k) s u c h t h a t f o r s(k) = t(n(k)) we have
} '/-) (U) - - ~J (S (~))1" ] v:tr (ll) ] dl,~ k=~ t \
(s(k))dh(u)) T
~ -4- ( ( m > s ( k + l ) _ < m ) ~(k))2-_}_
s(k)
s
F r o m (4.12) we then find t h a t [h (s (k-t- 1))--t~(s (k)) I < oo. h=l
T h e r e f o r e , h(s(k)) f o r k ~ oo h a s a s t r o n g l i m i t . Since s(k) ~ t , it f o l l o w s t h a t h(s(k)) c o n v e r g e s w e a k l y to ~a(t). T h u s , h(s(k)) ~ tTl(t) s t r o n g l y in H, a n d , s u b s t i t u t i n g the n u m b e r s s(k) in p l a c e of t in (4.13), a s k ~ o0 we o b t a i n (4.13) f o r the t c h o s e n . T h i s c o m p l e t e s the p r o o f of T h e o r e m 3.1. CHAPTER ITO
STOCHASTIC
EQUATIONS
METHOD 1.
OF
II IN B A N A C H
SPACES.
MONOTONICITY
Introduction In t h i s c h a p t e r we c o n s i d e r the Ito e q u a t i o n s t
t
,(t)=u0+ j'A(~(s), s)ds+ j'B(~(s), s)aw(s) 0
(1.1)
0
in B a n a c h s p a c e s . The c o e f f i c i e n t s A ( v , s), B(v, s) (of " d r i f t " and " d i f f u s i o n " ) a r e g e n e r a l l y a s s u m e d to be unbounded, nonlinear operators. T h e y m a y d e p e n d on the e v e n t in n o n a n t i e i p a t o r y f a s h i o n . By w we u n d e r s t a n d a W i e n e r p r o c e s s with v a l u e s in s o m e H i l b e r t s p a c e . An e x i s t e n c e and u n i q u e n e s s t h e o r e m w i l l be p r o v e d f o r a s o l u t i o n of an e q u a t i o n s l i g h t l y m o r e g e n e r a [ t h a n (1.1), and c e r t a i n q u a l i t a t i v e r e s u l t s on the s o l u t i o n wit[ be o b t a i n e d . A s o l u t i o n is u n d e r s t o o d to be a t r a j e c t o r y with v a l u e s in the d o m a i n of the o p e r a t o r s A ( . , t), B ( . , t) (which d o e s n o t d e p e n d on t) t h a t s a t i s f i e s (1.1) and is c o n s i s t e n t with the s a m e s y s t e m of i t - a l g e b r a s a s w(t), A ( . , t), and B ( . , t). T h i s s y s t e m i s a s s u m e d to be given a s w e l l a s the o r i g i n a l p r o b a b i l i t y s p a c e and the W i e n e r p r o c e s s . A s o l u t i o n is t h u s u n d e r s t o o d in the " s t r o n g " s e n s e . The m a i n c o n d i t i o n s on A a n d B a r e the c o n d i t i o n s of m o n o t o n i e i t y and c o e r c i v e n e s s [see (A2), (A3) in S e e . 2 of t h i s c h a p t e r ] . The f o l l o w i n g e q u a t i o n s s a t i s f y t h e s e a s s u m p t i o n s in s p a c e s of S o b o l e v t y p e : om
0 rn
.[p--2 t} rn
du(t, x) = a(t, co)(--1)~+t0--~( U2-m u(t, X~l
p>l,
xGG~R I, --2(p--1)a+4-1p~b2~<--s, n
du(t, x) =l, =1
1250
~
,.
P
u t~, x)) at -}- b (t, co) - ~ u (t, x) 2dw (t).
~>0, andw(t) is a Wienerproeess with values in RI; m
it
]=l t=1
~
(1.2)
x E R n, aij, bij a r e b o u n d e d , m e a s u r a b l e f u n c t i o n s s u c h tLaat f o r s o m e X > 0
t,]=I I=l
i,/=l
i=l
f o r a l l t, x, ~v f o r any v e c t o r ~ E Rn; wi(t) a r e i n d e p e n d e n t W i e n e r p r o c e s s e s with v a l u e s in R '1, and co is an "event."
T h e s e and o t h e r s t o c h a s t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s a r e c o n s i d e r e d in d e t a i l in Chap. III. The r e s u l t s of the p r e s e n t c h a p t e r a r e a r e f i n e m e n t of the r e s u l t s of P a r d o u x [84, 83] w h i c h , in t u r n , g e n e r a l i z e the r e s u l t s of B e n s o u s s a n and T e m a m [46]. A s a l r e a d y m e n t i o n e d a b o v e , we have s u c c e e d e d in s h o w i n g that c e r t a i n c o n d i t i o n s of P a r d o u x a r e s u p e r f l u o u s , in p a r t i c u l a r , the Local L i p s c h i t z c o n d i t i o n f o r the o p e r a t o r B. The m e t h o d of p r o v i n g the e x i s t e n c e t h e o r e m (the m o s t d i f f i c u l t and i m p o r t a n t p a r t of t h i s c h a p t e r ) h a s b e e n b o r r o w e d f r o m P a r d o u x and c o r r e s p o n d s to a G a l e r k i n s c h e m e : a f i n i t e - d i m e n s i o n a l a n a l o g of Eq. (1.1) is c o n s i d e r e d (Sec. 3), e s t i m a t e s of the s o l u t i o n i n d e p e n d e n t of the d i m e n s i o n a r e o b t a i n e d (See. 4), and then (by the m e t h o d of m o n o t o n i c i t y ) a p a s s a g e to the l i m i t is r e a l i z e d (See. 5). The b a s i c i m p r o v e m e n t s w h i c h m a k e it p o s s i b l e in the final a n a l y s i s to g e n e r a l i z e the r e s u l t s of P a r d o u x a r e m a d e at the f i r s t s t e p in S e e . 3. H e r e a t h e o r e m i s o b t a i n e d which g e n e r a l i z e s the well-~mown t h e o r e m of [to on the e x i s t e n c e and u n i q u e n e s s of s t r o n g s o l u t i o n s of a s t o c h a s t i c e q u a t i o n with r a n d o m c o e f f i c i e n t s s a t i s f y i n g L i p s c h i t z c o n d i t i o n s . 2.
Assumptions.
Formulation
of the
Main
Results
L e t ( ~ , :g', P) be a c o m p l e t e p r o b a b i l i t y s p a c e with an e x p a n d i n g s y s t e m of o - - a l g e b r a s {~-~} (lE[0, T], T < ~ ) , i m b e d d e d in ~ . We s h a l l a s s u m e that the f a t u i t y {~'t} h a s b e e n c o m p l e t e d with r e s p e c t to the m e a s u r e P. F u r t h e r , l e t H and E be r e a l , s e p a r a b l e H i l b e r t s p a c e s , w h e r e b y H and E a r e n a t u r a l l y i d e n t i f i e d with t h e i r d u a l s H* and E*; let w(t) be a W i e n e r p r o c e s s in E with n u c l e a r e o v a r i a n c e o p e r a t o r Q (see Chap. I, S e e . 2), and l e t z(t) be a s q u a r e - i n t e g r a b l e m a r t i n g a l e in H. We a l s o c o n s i d e r a r e a l , s e p a r a b l e , r e f l e x i v e B a n a e h s p a c e V and i t s dual s p a c e V*. As in Chap. I, if v i s an e l e m e n t of V and v* an e l e m e n t of V*, then vv* d e n o t e s the value of v* on v. I ' I x and (. , . ) X d e n o t e , r e s p e c t i v e l y , t h e n o r m in the s p a c e X and the s c a l a r p r o d u c t in the s p a c e X if X is a H i l b e r t s p a c e . In S e e . 3, w h e r e f i n i t e - d i m e n s i o n a l s p a c e s a r e c o n s i d e r e d , t h i s n o t a t i o n is s i m p l i f i e d ; s p e c i a l m e n t i o n is m a d e of t h i s .
.~Q(E, H), as b e f o r e , is the s p a c e of a l l l i n e a r o p e r a t o r s 9 d e f i n e d on Q~/2E and t a k i n g QI/2E into H s u c h that @Q~/2~cS2(E, it)(the s p a c e of H i l b e r t - S c h m i d t o p e r a t o r s f r o m E to H ) . . ~ Q ( E , H) is a s e p a r a b l e H i l b e r t s p a c e r e l a t i v e to the s c a l a r p r o d u c t (4>, ~)Q = t r 4,Q1/2(pQ1/2) *. The n o r m in this s p a c e we denote by I. tQ. The f o l l o w i n g a s s u m p t i o n s a r e h e n c e f o r t h u s e d : a) VcH=-H*~V*; b) V is d e n s e in H (in the n o r m of H); c) t h e r e e x i s t s a c o n s t a n t e s u c h that f o r a l l v e V, I vl H <- c i v I v ; d) vv* - (v, v*) H if v* E H. An i m p o r t a n t e x a m p l e of s p a c e s p o s s e s s i n g p r o p e r t i e s a ) - d ) i s t h e Sobolev s p a c e W p ( G ) ( = V ) a n d L2(G)(=H), w h e r e G is a b o u n d e d d o m a i n in R d, and dp >- 2(d - m p ) . T h e s e s p a c e s a r e d i s c u s s e d in m o r e d e t a i l in Chap. HI. S o m e o t h e r t r i p l e s of s p a c e s p o s s e s s i n g p r o p e r t i e s a ) - d ) a r e a l s o p r e s e n t e d t h e r e . We r e c a l l a l s o t h a t in Chap. I t r i p l e s of s p a c e s V, H, V ' c o n n e c t e d b y l e s s r i g i d a s s u m p t i o n s have a l r e a d y b e e n c o n s i d e r e d , and the " i m p l i c a t i o n " of a s s u m p t i o n s a ) - d ) w a s d i s c u s s e d in s o m e d e t a i l ; in p a r t i c u l a r , the p o s s i b i l i t y of i d e n t i f y i n g V ' with V* by m e a n s of (" , . ) H w a s d i s c u s s e d . We fix n u m b e r s p a n d q ,
pE (1,~), q
p/(p-
1).
S u p p o s e that f o r e a c h (v, t, co) e V x [0, T] x t2
J(v, t, oJ)~Y*, B(v, t,
o))e.~q(E, H).
We a s s u m e that f o r e a c h v e V the f u n c t i o n s A(v, t, co), B(v, t, w) a r e ( L e b e s g u e ) m e a s u r a b l e in (t, w) ( r e l a t i v e to the m e a s u r e dt x dP) and a r e & r - , - c o n s i s t e n t , i . e . , f o r e a c h v e V, t e [0, T] they a r e I V ' t - m e a s u r a b l e in w.
1251
We r e c a l l t h a t , s i n c e V* and .~@(E, /-/) a r e s e p a r a b l e , the c o n c e p t s of s t r o n g and w e a k m e a s u r a b i l i t y c o i n c i d e , and we s h a l l s p e a k s i m p l y of m e a s u r a b i l i t y . S u p p o s e f u r t h e r that on ~2 t h e r e is given anY-0 - m e a s u r a b l e f u n c tion u 0 with v a l u e s in H, w h i l e on [0, T] x f2 t h e r e is g i v e n a n o n n e g a t i v e function f(t, co) m e a s u r a b l e in (t, co) and :g't - c o n s i s t e n t . We a s s u m e t h a t f o r s o m e c o n s t a n t s K, a > 0 and f o r a l l v, v l , v2 E V, (t, co) ~ [0, T] • f~ the f o l l o w i n g c o n ditions are satisfied: A 1) 8 e m i c o n t i n u i t y of A : the function vA (v~ + Xv 2) i s c o n t i n u o u s in X on R i . A2) M o n o t o n i e i t y of (A, B): 2 (% - - v2) (A (%) - - A (v2)) + [ B (%) - - B (v2) [ 2~~
2vA(v)+lB(v)l~ +a]vlP-.< f +Klvl~. A 4) B o u n d e d n e s s of the g r o w t h of A: z
T
As)
MlUo]~< ~,
MSf(t) d t < m . 0
U n d e r t h e s e a s s u m p t i o n s we c o n s i d e r on [0, T] • f~ the s t o c h a s t i c e v o l u t i o n e q u a t i o n t
v ( t , co)=u0((o)+ i
t
A(v(s, o)), s, o))ds @fB(v(s,~
0
o)), s,
o)) dw (s,
o))+z(t, o)).
(2.1)
0
D e f i n i t i o n 2.1. A s o l u t i o n (or a V - s o l u t i o n ) of Eq. (2.1) is a function v(t, co) with v a l u e s in V d e f i n e d on [0, T] • f~, m e a s u r a b l e in (t, co), 9rt - c o n s i s t e n t , and s a t i s f y i n g the i n e q u a l i t y T
M I ,) ( I v ( f ) ] ~ + ] v ( t ) l ~ ) d t < ~
(2.2)
O
and Eq. (2.1) in the s e n s e of e q u a l i t y of e l e m e n t s of V* f o r a l m o s t a l l (t, co) ~ [0, T] • f~. In t h i s d e f i n i t i o n it is i m p l i c i t l y a s s u m e d that i n t e g r a l s (2.1) and (2.2) a r e m e a n i n g f u l . i s , h o w e v e r , s u p e r f l u o u s in view of w h a t f o l l o w s .
This assumption
By L e m m a 1.2,1 f,~.ction v is m e a s u r a b l e not only a s a function with v a l u e s in V but a l s o a s a function with v a l u e s in H. T h e r e f o r e , I v ( t ) l v , Iv(t)lH a r e m e a s u r a b l e with r e s p e c t to (t, w), and c o n d i t i o n (2.2) is meaningful. We show f u r t h e r t h a t the f u n c t i o n s A (v(t), t), B (v(t), t) a r e m e a s u r a b l e in (t, co) and ~ - t - m e a s u r a b l e in co. We f i r s t c o n s i d e r A(v(t), t). We u s e t h a t s a m e s o r t of l e m m a a s in the t h e o r y of m o n o t o n e o p e r a t o r s (see, e . g . , [30]). L E M M A 2.1.
If a s e q u e n c e v n e V c o n v e r g e s s t r o n g l y in V to v, then A (vn) c o n v e r g e s w e a k l y to A (v).
P r o o f . B e c a u s e of a s s u m p t i o n (A4), f o r any s u b s e q u e n e e {p} of n a t u r a l n u m b e r s the s e q u e n c e A(v#,) is b o u n d e d in V*, and t h e r e f o r e t h e r e e x i s t s a s u b s e q u e n c e {v} of the s e q u e n c e {g} a l o n g w h i c h A (vu) c o n v e r g e s w e a k l y to s o m e A ~ . We s h a l l show t h a t A ~ = A (v). L e t u b e an a r b i t r a r y e l e m e n t of V.
By a s s u m p t i o n CA2),
( a - - % ) (A ( a ) - A ( % ) ) - - K i u - - v , i ~-.< 0. P a s s i n g to the l i m i t w ~ co in t h i s e q u a l i t y , we o b t a i n (u-- v) (A (u) - - A m ) - - K ] u - -
v 13 "-< O.
(2.3)
We now s u p p o s e t h a t u = v + k y , w h e r e X ~ R+, and y is s o m e e l e m e n t of V; f r o m (2.3) it then f o l l o w s that
u (A (v §
1252
) --K~ i y I ~--
P a s s i n g to the l i m i t X i 0 in t h i s e q u a l i t y and u s i n g the s e m i c o n t i n u i t y of A [ a s s u m p t i o n (At)], w e o b t a i n g (A (v) - - A~) -.< 0. In view of the f a c t t h a t y is a r b i t r a r y , f r o m t h i s it f o l l o w s t h a t Aoo = A(v), a n d , s i n c e the s u b s e q u e n c e ~ } w a s a r b i t r a r y , the p r o o f of the [ e m m a i s c o m p l e t e . It f o l l o w s f r o m the [ e m m a t h a t the function v ~ uA(v, t, co) i s c o n t i n u o u s f o r e a c h u e V , (t, co) ~ [0, T] x ~2o T h u s , A (v(t), t , co) is w e a k l y , and h e n c e s t r o n g l y , m e a s u r a b l e in (t, co) and is J % - c o n s i s t e n t t o g e t h e r with v(t). M o r e o v e r , in view of (A4), (As), A ( v ( t , co), t , co) i s a s u m m a b l e function of t f o r a l m o s t a l l co if (2.2) is satisfied. The p r o b l e m of the m e a s u r a b i l i t y of B(v(t, co), t , w) is s o l v e d m o r e s i m p l y , s i n c e it is o b v i o u s t h a t by a s s u m p t i o n s (Az) and (A4), B ( . , t, co) f o r a l l t, co is a s t r o n g l y c o n t i n u o u s function f r o m V to ~ Q i E , ft). F r o m a s s u m p t i o n s (A3), (A 4) it f o l l o w s t h a t f o r a l l u, t, co
IB(., t, ~ ) i ~ < c (/(t, 0,)+i.l ~. -r-1 ' tt ] [)-
(2.4)
Therefore, T
MfIB(v(t), t)i~d
< oo,
o
the s t o c h a s t i c i n t e g r a l in (2.1) is d e f i n e d , and is a s q u a r e - i n t e g r a b [ e m a r t i n g a l e in H. R e m a r k 2.1. It is o b v i o u s (see Chap. [, S e e . 2, c l a r i f i c a t i o n s of T h e o r e m 2.6) t h a t v is a s o l u t i o n of (2.1) if and only if f o r s o m e s e t Y ~ V d e n s e in V (in the n o r m of V) f o r any u ~ Y t
l
(., ~ (t)).= (~, . o ) . + i ua (~ (~), s) ds + f ?~ (~ (s), s) dw (s)+ (., ._ (t)).. o
~. ~.
(t, ~o)).
o
D e f i n i t i o n 2.2. An H - s o l u t i o n of Eq. (2.1) is a s o l u t i o n u(t, co) with v a l u e s in H d e f i n e d on [0, T] x f2, s t r o n g l y c o n t i n u o u s in H w i t h r e s p e c t to t, Y t - c o n s i s t e n t , and s u c h t h a t l) u e V
[ a . e . (t, ~)1 t
t/l J (]u(t) l S + l u ( t ) [ ~ ) d t < c-o; o 2) t h e r e e x i s t s a s e t 2 ' ~ 2
of t o t a l p r o b a b i l i t y on w h i c h f o r a l l t e
u ( t ) - tto +
[% TI
i A (u (s), s) ds + [ B (u (s), s) dw (s) + z (t),
(2.5)
w h e r e the e q u a l i t y is u n d e r s t o o d a s an e q u a l i t y of e l e m e n t s of V*. In c l a r i f i c a t i o n of (2.5) we note that by L e m m a 1.2.1 XV(U(t)) is m e a s u r a b l e in (t, co), 5t-t - m e a s u r a b l e co, e q u a l to 1 [ a . e . (t, co)] b y the f i r s t r e q u i r e m e n t , and the i n t e g r a l s in (2.5) a r e t h e r e f o r e m e a n i n g f u l a s
i ~v (u (s)) A (t~(s), s) ds,
o
D e f i n i t i o n 2.3.
in
j' xv (~ (s)) B (it (s), s) dw (s). o
An H - s o l u t i o n u is a c o n t i n u o u s m o d i f i c a t i o n of a s o l u t i o n v in H if u(t, w) = v(t, a') [aoe.
(t, co)l. We now f o r m u l a t e the m a i n r e s u l t s of t h i s c h a p t e r .
A l l the a s s u m p t i o n s a b o v e a r e a s s u m e d to b e s a t i s -
fied. T H E O R E M 2.1.
A s o l u t i o n v of Eq. (2.1) e x i s t s .
F r o m t h i s t h e o r e m a n d T h e o r e m 1.3,2 we i m m e d i a t e l y o b t a i n the f o l l o w i n g r e s u l t . C O R O L L A R Y 2.1.
T h e r e e x i s t s a c o n t i n u o u s m o d i f i c a t i o n of the s o l u t i o n v of Eq. (2.1) in H.
T h e o r e m 2.1 is p r o v e d in S e c . 5. Eq. (2.1).
The n e x t t h e o r e m c o n t a i n s the u n i q u e n e s s a s s e r t i o n f o r the s o l u t i o n of
T H E O R E M 2.2. L e t vn(t), n 0, 1, 2 . . . . . b e s o l u t i o n s of Eq. (2.1) with i n i t i a l d a t a u 0 u n, w h e r e M lu~l~i < ~ m~d M lu~ - u~l~t - - 0 a s n - - ~ . S u p p o s e t h a t un(t) a r e c o n t i n u o u s m o d i f i c a t i o n s of ~ ( t ) in H. Then f o r any e > 0
1253
lira {sup M] u '~(t)--uO(t)I~} q - P {sup [ tL=(t)-uo(t)[~> q = O R e m a r k 2.2. We s h a l l s e e f r o m the p r o o f of the t h e o r e m t h a t it is v a l i d if in the d e f i n i t i o n of a s o l u t i o n c o n d i t i o n (2.2) i s d r o p p e d while only c o n d i t i o n (A 2) is r e q u i r e d of the c o e f f i c i e n t s of (2.1). T H E O R E M 2.3.
If v is a s o l u t i o n of Eq. (2.1) and u i s i t s c o n t i n u o u s m o d i f i c a t i o n in H, then
Iv(t)l~,dtKc
Msuplu(t)l~+-a'ly t..~ T
0
(
Mltto[Sq-M
;
)
f(t)dtq-Mlz(T)] 5 ,
0
w h e r e c d e p e n d s o n l y on K, p, T, and ~ . T h e o r e m s 2.2 and 2.3 a r e p r o v e d in S e e . 4. The f o l l o w i n g t h e o r e m on the M a r k o v p r o p e r t y of s o l u t i o n s of (2.1) a l s o b e l o n g s to the b a s i c r e s u l t s of the c h a p t e r . T h i s t h e o r e m is p r o v e d at the e n d of S e e . 5. T H E O R E M 2.4. S u p p o s e t h a t A and B do not d e p e n d on w, z(t) -= 0, v(t) is a s o l u t i o n of Eq. (2.1), and u is i t s c o n t i n u o u s m o d i f i c a t i o n in H; then u(t) is a M a r k o v r a n d o m v a r i a b l e . R e m a r k 2.3. If in c o n d i t i o n s (A 2) and (A 3) we I n d e e d , if v i s a s o l u t i o n of (2.1), then v(t)e - K t is a e - K t ( A - KI), w h e r e I i s the i d e n t i t y o p e r a t o r , and the a r g u m e n t s of A and B a r e a l s o c h a n g e d , then it and (A 3) with K = 0.
w e r e to t a k e K = 0 t h i s would o c c a s i o n no l o s s of g e n e r a l i t y . s o l u t i o n of an e q u a t i o n of t y p e (2.1) with A r e p l a c e d b y with B r e p l a c e d b y e - K t B . If it is f u r t h e r c o n s i d e r e d that is e a s y to show t h a t the new A and B s a t i s f y c o n d i t i o n s (A 2)
R e m a r k 2.4. O u r a s s m n p t i o n t h a t the s p a c e s in q u e s t i o n a r e r e a l is not e s s e n t i a l . It m a y be r e l a x e d if in c o n d i t i o n s (A1)-(A 3) in p l a c e of vA(v 1 + Xv2), (v 1 - v2)(A(vl) - A(v2)), vA(v) we w r i t e Re vA(v 1 + ?~v2), Re (v 1 v 2) (A (v 1) - A (v2)) , Re vA(v). 3.
Ito
Equations
in
Rd
L e t R d be E u c l i d e a n s p a c e of d i m e n s i o n d with a f i x e d o r t h o n o r i n a [ b a s i s , let x i be the i - t h c o o r d i n a t e of a point x ~ R d, let (fl,:~'-,P)be a c o m p l e t e p r o b a b i l i t y s p a c e , and l e t {~'t}, t >- 0, be an e x p a n d i n g f a m i l y of c o m p l e t e c - a l g e b r a s : g - t c ~ ' . L e t In(t) b e a d l - d i m e n s i o n a l , c o n t i n u o u s , l o c a l m a r t i n g a l e r e l a t i v e to {~-~}, with in(0) = 0, and l e t Aft) be a c o n t i n u o u s , r e a l , n o n d e c r e a s i n g f t - c o n s i s t e n t p r o c e s s with A 0 = 0. Suppose that f o r t -> 0, x ~ R d, w ~ ~2 a d x d 1 m a t r i x b ( t , x) and a d - d i i n e n s i o n a [ v e c t o r a f t , x) a r e d e f i n e d . We a s s u m e t h a t f o r e a c h x ff R d, a ( t , x) and b ( t , x) a r e c o m p l e t e l y m e a s u r a b l e r e l a t i v e to {~z-t} and a r e c o n t i n u o u s in x f o r e a c h (t, w). L e t x 0 be a d - d i i n e n s i o n a l J - o - m e a s u r a b l e q u a n t i t y . We c o n s i d e r the f o l l o w i n g e q u a t i o n : t
t
x (t) ~ x o+ ~ a (s, x (s)) dA (s) + ~ b (s, x (s)) dm (s). 0
(3.1)
0
E q u a t i o n (3.1) w i l l be c o n s i d e r e d u n d e r c e r t a i n a d d i t i o n a l c o n d i t i o n s on a , b , A, and m whose f o r m u l a tion r e q u i r e s the f o l l o w i n g n o t a t i o n . By the D o o b - M e y e r t h e o r e m t h e r e e x i s t s a c o n t i n u o u s , i n c r e a s i n g p r o c e s s d e n o t e d b y ( I n } t f o r w h i c h (m2(t) - (In}t) is a l o c a l m a r t i n g a l e [ r e l a t i v e to{~t't}], and (m}0 = 0. F o r i, j 1, . . . . d 1 we f u r t h e r d e f i n e b y m e a n s of the D o o b - M e y e r t h e o r e m c o n t i n u o u s p r o c e s s e s ( i n i, mJ }t h a v i n g l o c a l l y b o u n d e d v a r i a t i o n in t f o r w h i c h (mi(t)in j (t) - (In i, InJ }t i s a l o c a l I n a r t i n g a l e and ( m i , mJ }0 - 0. We r e c a l l t h a t the m a t r i x ({In i, mJ }t ) is n o n n e g a t i v e d e f i n i t e and dt
(m)t~- x
(rni, m ~ ) t
i=I
f o r a l l t (a. s . ) . We fix a c o n t i n u o u s , r e a l , n o n d e c r e a s i n g , ~-, - c o n s i s t e n t p r o c e s s Vt with V 0 - 0 such t h a t f o r e a c h w the m e a s u r e s on the t a x i s g e n e r a t e d b y the f u n c t i o n s Aft), ( m } t a r e a b s o l u t e l y c o n t i n u o u s r e l a t i v e to the m e a s u r e c o r r e s p o n d i n g to V t (e. g . , V t - A (t) + ( i n } t ) . We set
c~;( t ) ~
a(t,x):a(t,
d ~ .~,dv~,,d > ~ '
dA(t) x)--dV 7 ,
C (t) = (c ,j (t)),
~(t, x)--b(t, x)CI/2.
We s h a l l a s s u m e t h a t the foUowing c o n d i t i o n s a r e s a t i s f i e d in a d d i t i o n to t h o s e e n u m e r a t e d a b o v e : e a c _ ~ x E R d, T > 0
1254
for
T
T
j~ la(t, x) l d V t = S la(t, x) l d A ( t ) < e~ 0
(a.s.);
(3.2)
O
for any R > 0 t h e r e e x i s t s a n o n n e g a t i v e , c o m p l e t e l y m e a s u r a b l e p r o c e s s Kt(R) such that T
f K (1~) dV t < oo
(a. s.);
0
and for all T >- 0 mad for e a c h z, x, y E R d such that Ixl, lyl -< R f o r a l m o s t all t r e l a t i v e to the m e a s u r e dV t
2 (x - v) (~ (t, x) - ~ (t, y)) +it [~ (t, x) - [ ~ (t, v)[l s < -,% (t?) (x - v) ~,
2za (t, z) -~ II ~ (t, z)i]2-.
(1 +z2),
(3.3) (3.4)
w h e r e K t = Kt(1), ll7[I for a m a t r i x 7 m e a n s (tr 7"y*)1/~, and for e, 6 E R d, 5e d e n o t e s the s c a l a r p r o d u c t of 5, g. T H E O R E M 3.1. T h e r e e x i s t s a c o n t i n u o u s , S t ' t - c o n s i s t e n t p r o c e s s x(t) for which (3.1) is s a t i s f i e d for a l l t with p r o b a b i l i t y 1. If x(t), y(t) a r e two c o n t i n u o u s ~ t - c o n s i s t e n t p r o c e s s e s s a t i s f y i n g (3.1) for e a c h t (a. s.), then
P {sup l x ( t ) - y ( ~ ) i > 0} = 0 . t>~0
The p r e s e n t t h e o r e m is a g e n e r a l i z a t i o n of I t o ' s c l a s s i c a l t h e o r e m on the e x i s t e n c e of a s t r o n g s o l u t i o n of a s t o c h a s t i c e q u a t i o n of type (3.1) with r a n d o m c o e f f i c i e n t s , rn the p r e s e n t t h e o r e m we have a v o i d e d the L i p s c h i t z c o n d i t i o n and r e p l a c e d it by c o n d i t i o n (3.3) which we s h a l l c a l l the m o n o t o n i e i t y c o n d i t i o n . We p r e s e n t an e x a m p l e of an e q u a t i o n with c o e f f i c i e n t s which s a t i s f y o u r c o n d i t i o n s but do n o t s a t i s f y a L ipsehttz condition. E x a m p l e . Let w(t) be a o n e - d i m e n s i o n a l W i e n e r p r o c e s s , tel p E (1, c o m p l e t e l y m e a s u r a b l e f u n c t i o n s s u c h that
2),
and let aft, co) and b(t, co) be
T
,i" a (t, o~) dt < ~
(a. s.)
0
and p2
- - 2 ( p - - 1 ) a ( t , co)@-~- b2(t, co)~
((t,r
a.e.)
We c o n s i d e r the e q u a t i o n d x (t) = --l x (t) ?-' sgn x (t) c~ (t) d t + l x (t) J"~b (t) dw (t).
It is c l e a r that for p < 2 the c o e f f i c i e n t s of t h i s e q u a t i o n do not s a t i s f y L i p s c h i t z c o n d i t i o n s , but the m o n o t o n i c i t y c o n d i t i o n is s a t i s f i e d . Indeed, by the f o r m u l a of H a d a m a r d i
- 2a (1 x l"-' sgn x - i y
Ip-' sgn ~) ( x -
y) + ~ (I x Ip''~ - ! ~ IP~)~ = - 2a ( x -
v) ~ j' ( p -
1)I x + 9 ( v - x)!p-~dT
o / 1
)2
1 0
K we take V t - t, then f r o m t h i s i n e q u a l i t y we o b t a i n the m o n o t o n i e i t y c o n d i t i o n with Kt - 0 and a l s o c o n dition (3.4), s I n c e in the p r e s e n t c a s e c~(t, 0) - [3(t, 0) = 0. T h u s , c o n d i t i o n (3.3) is w e a k e r than the L i p s c h i t z c o n d i t i o n . H o w e v e r , e x t r e m e l y "bad" f u n c t i o n s m a y n o t s a t i s f y (3.3). We r e m a r k without p r o o f that (3.3) i m p l i e s the d i f f e r e n t i a b i l i t y of c~(t, x) with r e s p e c t to x f o r a l m o s t all x and that the f i r s t g e n e r a l i z e d d e r i v a t i v e s of B(t, x) with r e s p e c t to x a r e l o c a l l y s q u a r e - s u m mabte. We note that In c o n t r a s t to I t o ' s w o r k we c o n s i d e r the e q u a t i o n o v e r a s e m i m a r t i n g a l e . G e n e r a l i z a t i o n s of I t o ' s r e s u l t s to the c a s e of i n t e g r a t i o n o v e r a s e m i m a r t i n g a l e have b e e n o b t a i n e d by K a z a m a k i [65], D o t 6 a n s Dade [58], P r o t t e r [86], G a l ' e h u k {16], L e b e d e v [29], and o t h e r s . The r e s u l t s of t h e s e w o r k s a r e a s p e c i a l
1255
c a s e of o u r a s s e r t i o n a s f a r a s c o n c e r n s c o n t i n u o u s s e m i m a r t i n g a l e s . It s e e m s to us that the e x t e n s i o n of T h e o r e m 3.1 to the c a s e of d i s c o n t i n u o u s s e r e [ m a r t i n g a l e s i s an a l t o g e t h e r a c c e s s i b l e p r o b l e m , but it is b e y o n d our present interests. A s a l w a y s in s i m i l a r s i t u a t i o n s , the u n i q u e n e s s a s s e r t i o n in T h e o r e m 3.1 is v e r y s i m p l e . In o r d e r to p r o v e it, we r e m a r k f i r s t t h a t b y the c o n t i n u i t y of x ( t ) , y ( t ) , a n d the i n t e g r a l s c o n t a i n e d in (3.1) with r e s p e c t to t the p r o c e s s e s x ( t ) , y(t) s a t i s f y Eq. (3.1) f o r a l l t with p r o b a b i l i t y 1. F u r t h e r , we s e t %(R)=exp
--
K,(R) dV, .
We a p p l y I t o ' s f o r m u l a to Ix(t) - y(t)12~(R) and u s e (3.3).
We then find t h a t ( a . s . ) f o r a l l t
tA~(R)
I x (t A r (R))--Y (t A v (R))12tP, A,(R)--< 2 ~
% (R) (x (s)--V (s)) (b (s, x (s))--b (s, Y (s))dm (s)----m;(R),
0
w h e r e r(R) is the t i m e of the f i r s t e x i t of I x (t)IV] V (t) l f r o m [0, R). We s e e that the l o c a l m a r t i n g a l e m~(R) is n o n n e g a t i v e , a n d h e n c e m~(R) is a s u p e r m a r t i n g a l e . S i n c e m~(R) = 0, it f o l l o w s t h a t m~(R) = 0 (a. s . ) , I x (t/~v (R))-- Y (t A r (R)) 1= 0 (a.s.) f o r a n y t , and, s i n c e I x(t) - y (t)l is c o n t i n u o u s in t and R is a r b i t r a r y , it f o l lows t h a t sup I x ( t ) - : y (t) [ = 0 (a. s.) a s r e q u i r e d . t
The e x i s t e n c e a s s e r t i o n in T h e o r e m 3.1 w i l l be p r o v e d a f t e r p r o v i n g a n u m b e r of a u x i l i a r y p r o p o s i t i o n s . In T h e o r e m 3.1 it is a s s e r t e d , in p a r t i c u l a r , t h a t the r i g h t s i d e of (3.1) e x i s t s f o r s o m e p r o c e s s x(t). T h i s f a c t f o r any b% - c o n s i s t e n t , c o n t i n u o u s p r o c e s s f o l l o w s i m m e d i a t e l y f r o m the f o l l o w i n g t e m m a . L E M M A 3.1.
F o r any T , R > 0 ( a . s . ) T
F
I s.p
It, x), A It/<
[ sup jj It, xt
dV t <
oO.
P r o o f . L e t ~ryi} be a c o u n t a b l e , d e n s e s u b s e t o f { x : l x l -< R + 2}. We s e t (Dn(~, x) = max(c~(x - Yi), i = 1 . . . . . n}, c~, x ~ R d. It is c l e a r t h a t ~n(C~, x) a r e c o n t i n u o u s f u n c t i o n s and ~n(C~, x) r ~ x + Ic~l(R + 2) a s n ~ ~ . The l a s t function is c o n t i n u o u s a n d g r e a t e r than two if Ix[ - R, I~1 1. By D i n i ' s t h e o r e m t h e r e e x i s t s an n s u c h that ~0n(C~, x) -> l f o r I x l - < R , I o z l = l . T h i s i m p l i e s t h a t ~0n(OZ, x) >_ l o w l i e r I x l - < t l a n d a t l c ~ e R d. Subs t i t u t i n g the p o i n t s Yi into (3.3) in p l a c e of y and c o m p u t i n g the u p p e r b o u n d s on i f o r Ixl -< R, we find that rz
2 ]a(t, x ) ] ~ < 4 K t ( R + 2 ) ( R + 2 ) ~ - 4 - 2 ( R
i-2) ,"~ [a(t, tq)~i" ~ l--I
B e c a u s e o f (3.2), t h i s p r o v e s the f i r s t a s s e r t i o n of the t e m m a . The s e c o n d a s s e r t i o n f o l l o w s in an o b v i o u s w a y f r o m the f i r s t and f r o m (3.4). The p r o o f of the l e m m a i s c o m p l e t e . L E M M A 3.2. L e t f(x) be a r e a l , l o c a l l y b o u n d e d function on R d, let n > 0, and let N = s u p { l f ( x ) l : [xl -< n}. Then on R d t h e r e e x i s t s a r e a l function g(x) s u c h t h a t g(x) f(x) f o r Ix[ -< n, g(x) : 0 f o r Ixl >- n + 1, Ig(x) l -< I f(x) l, and I g (x) - g (~) i~ < i f (x) - f (v) i" + N ~ (x - - ~)~
(3.5)
f o r a l l x, y ~ R d. M o r e o v e r , if f(x) i s c o n t i n u o u s in x and d e p e n d s in a m e a s u r a b l e way on s e v e r a l p a r a m e t e r s , then g(x) is c o n t i n u o u s in x a n d is m e a s u r a b l e with r e s p e c t to t h e s e p a r a m e t e r s . Proof.
We s e t h ( x ) - N ( n + 1 ) -
Nlxl g/+) ( x ) = m a x ( m i n ( h (x), f +(x)), 0), g(_) (x) = max (rain (h (x), f _ (x)), 0), g ( x ) = g ( + ) ( x ) - g(_) (x). *
We s h a l l p r o v e t h a t g s a t i s f i e s at[ the r e q u i r e m e n t s . The l a s t a s s e r t i o n of the t e m m a i s o b v i o u s , s i n c e N is a m e a s u r a b l e function of t h o s e p a r a m e t e r s on which f d e p e n d s , while the c o n t i n u i t y of g f o l l o w s f r o m the c o n t i n u i t y of f and (3.5). F u r t h e r , g(+) _< f+, g(_) _< f_. T h e r e f o r e , g~: = g(:L), Ig |-< Ifl and g(x) h a s the s a m e sign a s f(x). M o r e o v e r , it is o b v i o u s t h a t h(x) ~ 0 f o r | x l _ > n + 1 . T h e r e f o r e , g = 0 f o r I x l ~ n + 1. Since h ( x ) - > N >- If(x)l f o r Ixl -< n , it f o t l o w s t h a t g(x) - f(x) f o r Ixl -< n. * The method
1256
of constructing
g was
indicated to us by A. D. Venttsel'.
It r e m a i n s to p r o v e (3.5). of g(x) and g(y), and
We fix points x, y.
If f(x) and fly) have d i f f e r e n t s i g n s , then the s a m e is true
[g ( x ) - g @ I=llg (x)i+[ g @ll <[I f(x)[-}-[ f @ i l =l f ( x ) - f @ I. We now c o n s i d e r the c a s e w h e r e f(x) and f(y) have the s a m e sign. Suppose, to be s p e c i f i c , that fix) >- 0, f(y) >_ 0; then g(x) = g+(x), g(y) = g+(y). Since the m o d u l u s of the d i f f e r e n c e of the u p p e r (tower) bounds does not e x e e e d the u p p e r (upper) bound of the m o d u l u s of the d i f f e r e n c e , it follows that I g (x) -- g (g) [ ..< i rain (h (x), f (x)) -- rain (h (g), f (g)) I~< max (I h (x) -- h (iS) [, [ f i x) -- f (g) I). This o b v i o u s l y i m p l i e s (3.5).
The p r o o f of the i e m m a is c o m p l e t e .
LEMMA 3.3 For any n > O there exist p r o e e s s e s ~(t, x), ~(t, x), Nt sueh that [~(t, x) e Rd, b(t, x) is a d x dl m a t r i x , N t is a r e a l p r o c e s s , ~, 1~, N a r e defined f o r at[ x e R d, t >- 0, co e S2, a r e continuous in x, a r e c o m p l e t e l y m e a s u r a b l e , ~(t, x) = a(t, x), t~(t, x) = b(t, x) f o r txl --- n, ~(t, x) -- 0, b(t, x) = 0 f o r lxl -> n + 3 f o r all t, t
I N~dV~ < ee
(a. s.),
(3.~)
and for all x, y e R d In(t, x) l@[[~(t, X) l]2 -J~ N t,
2(x-y)(~x(t, x)-~(t, v))+ll~(r x)-~(t, ~r 2 ~
(3.7) (a. e. d P X d V t ) ,
whe re
dA (t) ~ ( t ) ~ a ( t ) - ~ 7 - ~ , 6(t)=b(t)C',2(t). P r o o f . We fix n ~ 1. L e t T t be an o r t h o g o n a l d 1 x d~ m a t r i x , and let At be a diagonal m a t r i x of the s a m e d i m e n s i o n so that C t = T t A t T ~. it is well known that such m a t r i c e s e x i s t , and they can be c h o s e n to be c o m p l e t e l y m e a s u r a b l e . On the b a s i s of e a c h e l e m e n t of the m a t r i x b(t, x)T t for fixed n we c o n s t r u c t by m e a n s of L e m m a 3.2 e l e m e n t s which we a s s i g n to the m a t r i x b ' ( t , x). We multiply the l a t t e r on the r i g h t by T~ and den o t e [ h e r e s u l t by b(t, x) . Suppose f u r t h e r that the funetionT1 ~ C ~e 0 ( Rd) , ~ ( x ) = l f o r Ixl-<-n + 2 , ~ ( x ) = 0 f o r I x l - > n + 3 , 0 - < ~ _< 1. We s e t h ( t , x ) = a ( t , x ) ~ ( x ) . We shall prove that t h e r e e x i s t s a p r o c e s s Nt such that f o r ~, t~, N the a s s e r t i o n s of the [ e m m a hold. It is not h a r d to see that by L e m m a s 3.1 and 3.2 t h e r e e x i s t s a p r o c e s s N~~) s a t i s f y i n g condition (3.6) and s u c h that
lhif(t, x ) - g (t, v)/?=l[ g(t, x) r , - ? (t, y) r,)Ap~i1~ (3.8)
..< NI z) ( x - - g)2 + ]l (O (t, x) T t -- b (t, g) rt) A~/2l] 2 = N~ ~) (x -- g)2 + II]3(t, x) -- 13(t, g)112. M o r e o v e r , by L e m m a 3,1 t h e r e e x i s t s a p r o c e s s N~2) s a t i s f y i n g (3.6) and such that
2 (x-
v) (~ (t, x) - ~ (t, v)) = 2 ( x - y) a (t, x) (n ( x ) - n (v)) + 2 (x - ~) n (y) (~ (t, x ) - ~ (t, y)) < 2 (n (x)An (y)) ( x -
(3.9)
v) (~ (t, x ) - a (t, y ) ) + N I-~) (x --y)~.
F u r t h e r , by L e m m a 3.1 t h e r e e x i s t s a p r o c e s s Nt(3) s a t i s f y i n g (3.6) and such that 2 l~(t, x)I~-l]~(t, x)l]2~N~ 3).
(3.t0)
F i n a l l y , if Ixl, lyl < n + 2, then ~(y) = 1 and the s e c o n d inequality ha (3.7) is s a t i s f i e d by (3.8), (3.9), and ( 3 . 3 ~ w i t h N ~ N ( 9 +N(3) + K ( n + 3 ) . If Ixl, l y l - > n + 1 , t h e n ~ ( x ) =~(y) = 0 a n d (3.7) is s a t i s f i e d with N : N(2) + K ( n + 3). If one of the values of ] x i , I y l i s l e s s t h a n n + 1 , while the o t h e r is g r e a t e r t h a n n + 2 , then Ix - y [ -> 1, and (3.7) is s a t i s f i e d b y ( 3 , 9 ) , (3.10), and (3.3) with N = N(2) + N(3) + Kba + 3), since one of the values of ~(t, x), ~(t, y) is z e r o . T h u s , inequalities (3.7) a r e s a t i s f i e d with N = N(0 + N(2) + N(3) + K(n + 3), and the p r o o f of the [ e m m a is c o m p l e t e . T
LEMMA 3.4. Suppose t h e r e e x i s t s a c o m p l e t e l y m e a s u r a b l e p r o c e s s Nt _> 0 sueh that f N t d V l < ~ (a. s.) f o r all T > 0 and f o r all y, x 0
[~(t,x)I+l~(t,x)12
(a.e. dPXdV),
1257
21x-yll~(<
y)[+l[[3(< x ) - ~ ( t , y ) l l 2 < N t l x - v l
x)-~(t,
~
Then Eq. (3.1) h a s a unique s o l u t i o n . Proof.
We have a l r e a d y p r o v e d the u n i q u e n e s s of the s o l u t i o n of (3.1) a b o v e .
~t-----exp(--3iNflV~--,xo,), t
We s e t
x~
t
(3.11)
x ~ + ~ = X o + l a (s, xr (s)) dA (s) + I b (s, x~ (S)) dm (s), r > O . 0
0
U s i n g I t o ' s f o r m u l a f o r (xr+~(t) - xr(t))2~t, it is e a s y to show that f o r s o m e l o c a l m a r t i n g a l e m r ( t ) f o r r->l t
(x ~+~(t) - x~ (t))~,~ = I {2 ( x ,+' (s) - x ~ (s)) (~ (s, x " (s)) - ~ (s, x ' - ' (s))) 0 ~ - tl ~ (S, X r (S)) - - ~ (S, .)2r-1 (S)) []2 __ 3 N ~ (.x r+l (S) - - X r (S)) 2} * s d V s - ~ M r (t)
t
<-:l {i X'~I (s)-- x t (s) l Ns Jx' (s)-- x "-I (s) [ + N , I xr (s) -- x '-I (s)12- 3N~ (x '+i (s) - - # (s)) 2} *~dV~ + rn ~(t) 0 t
.< I {-} x,r r (s)- x,-,(s)j2-
(,)j2}
0
In the i n e q u a l i t y b e t w e e n the e x t r e m e t e r m s in t h i s c h a i n of i n e q u a l i t i e s we t a k e the i n t e g r a l of the n e g a t i v e e x p r e s s i o n to the left s i d e , a n d we then s u b s t i t u t e in p l a c e of t the e x p r e s s i o n t = ~ A ~ , w h e r e T, z i a r e M a r k o v t i m e s s u c h t h a t Ti r ~ and m~(ft A ~ ~) is a m a r t i n g a l e ; c o m p u t i n g the m a t h e m a t i c a l e x p e c t a t i o n s and l e t t i n g t ~ ~o, we c o n c l u d e t h a t M [ 5 i (x'+l (s)-- x ~(s))2Nsr
-I- 2 (x "+' ('r) - - x r (~)12*, < M 3 . (x" (s) -- x'-' (s))2N~pflV~, r > 1,
0
0
w h e r e (xr(T + 1) - x r ( r ) ) 2 ~ r is t a k e n e q u a l to z e r o on the s e t w h e r e T = oo. S i m i l a r l y , 3I (x x(~))2 ~p~+ 2 ~ (x 1(s)) 2 N,~J,dV~ < Mx~e-'X.! + M 0
N~dV~
< Mx2oe -rx.r +-~1 < oo,
0
w h e r e (xl(T))2~bT is a l s o t a k e n e q u a l to z e r o on the s e t w h e r e r = ~ . s o m e c o n s t a n t N' f o r a l l r , 7
F r o m t h e s e i n e q u a l i t i e s it f o l l o w s t h a t f o r
MI 0
F r o m t h i s f o r z = v(r)
inf{t : (xr+l(t) - xr(t))2$t -> r -4} we have 3 r
]2){S[lp(jcr+l(,)__Xr (t))21pt~F-4}
l e m m a the s e r i e s
2
Ix~+l(t)--x'(t)[~112
c o n v e r g e s u n i f o r m l y f o r a l m o s t a l l w on e a c h f i n i t e t i m e s e g m e n t to s o m e c o n t i n u o u s p r o c e s s x(t). the l i m i t in (3.11), we c o m p l e t e the p r o o f o f the l e m m a . L E M M A 3.5. We c h o o s e n > 0, ~, 1~, N f r o m L e m m a 3.3. it h a s a unique s o l u t i o n .
P a s s i n g to
If in Eq. (3.1) a, b a r e r e p l a c e d b y a , 1~, then
P r o o f . F o r r = 1, 2 . . . . we d e f i n e f u n c t i o n s ~ r t~r by c o n v o l u t i o n of (}, 1~ on x with a 6 - t y p e s e q u e n c e of i n f i n i t e l y d i f f e r e n t i a b i e , c o m p a c t l y s u p p o r t e d , n o n n e g a t i v e f u n c t i o n s of the f o r m ~(Jrx)J d . w h e r e Jr -> 0, J r o~ a s r ~ ~ . S i n c e f o r a n y r the f i r s t d e r i v a t i v e s w i t h r e s p e c t to x and the f u n c t i o n s fir, b r can b e b o u n d e d in
1258
t e r m s of the m a x i m a of l a l , I b l , b y the f i r s t i n e q u a l i t y of (3.7) the c o n d i t i o n s of L e m m a 3.4 a r e s a t i s f i e d f o r h r t~r. T h e r e f o r e , f o r any r t h e r e e x i s t s a unique s o l u t i o n of the e q u a t i o n t
r
x ~ (t) = x o + j" ~t~ (s, x t (s)) d A (s) + j" ~r (s, x' (s)) dot (s). 0
(3.12)
0
S i n c e fi(t, x) = 0, tg(t, x) = 0 f o r Ixl -> n + 3 and g is a c o m p a c t l y s u p p o r t e d f u n c t i o n , it m a y b e a s s u m e d with no l o s s of g e n e r a l i t y t h a t f i r ( [ , x ) = 0 , 1 ~ r ( t , x ) = 0 f o r l x l - > n + 4. Tm t h i s e a s e if I x 0 1 - < n + 4 , then the p r o c e s s x r ( t ) n e v e r [ e a v e s the s e t {Ixl <- n + 4}. If Ix01 >-n + 4, then x r ( t ) = x 0 f o r at[ t. T h i s i m p l i e s t h a t
sup sup l x ~ ( t ) - x0 [-,< 2n + 8 (m H.). l>/0
Further,
we c h o o s e Nt f r o m L e m m a 3.3. ] j ~ ( t , xr(t))]]2~<supllg'(t,
We s e t q~t--exp
(J)
--., N~dV~
X
(3.13)
r
We note t h a t b y (3.7) f o r c}r = 8 r ( d A / d V ) , } r = ~ r c 1 / 2 x)H2~<sup ilk(t, x){12~
we have
i ~ ( t , xr(t))f~
(3.14)
and p r o v e t h a t lira sup M] xr ( ~ ) - - x ~ ' ( ~ ) 1 2 ~ = 0 ,
(3.15)
w h e r e the s u p r e m u m is t a k e n o v e r a l l M a r k o v t i m e s ~ and (Xr(T) -- xP(T))2$T iS s e t e q u a l to z e r o on {T = m}. By I t o ' s f o r m u l a , f o r s o m e l o c a l m a r t i n g a l e m r ' P ( t )
d[ x, (t)-- x~ (t)? ~,~, = {2 (x, (t)-- x~ (t)) ( ~ (t, x ~ ( t ) ) - - ~ (t, x~ (t))) + 1[# (t, x ~ (t))-- ~" (t, xP (t))112- Art (X r (t)-- xp (t))2} ~)tdVt @ 2 (x r (t) - - x p (t)) (~r (t, X p ( t ) ) - - cZP([, XP ( t ) ) ) * t d V t
+ tl ~" (t, xP (t)) -- ~p (t, xp (t))112,ed~.Zt -/2 [~" (t, xp (t)) --~p (t, xp (t)), 6 ~ (t, x" (t)) - [ ~ (t, x~' (t))l , t d V t + dnz ~'p (t),
(3.16)
w h e r e f o r two d x d 1 m a t r i c e s [~t, cr2] is the s u m of a l l p r o d u c t s of the f o r m (r~J~i j . A p p l i c a t i o n of the s e c o n d i n e q u a l i t y in (3.7) and the C a u c h y - B u n y a k o v s k i i i n e q u a l i t y s h o w s that the e x p r e s s i o n in b r a c e s in (3.16) is n o n p o s i t i v e ( a . e . d P • dVt). M o r e o v e r , the f u n c t i o n s
(x~ ( t ) - x~ (t)) (~, (t, xp ( t ) ) - gp (t, x~ (t))) ,~ b y (3.13), (3.14) a r e boLmded in a b s o l u t e v a l u e b y the function 4(2n + 8)Nt$ t w h i c h i s s u m m a b l e with r e s p e c t to d P x d V t , and f o r a l l t , w t h e y t e n d to z e r o a s r , p ~ oo by the c o n t i n u i t y of the c o m p a e t l Y d s u p p o r t e d ~ ( t , x) in x. F o r a t l t , cowehave Ic~r(t,x)-t}(t,x)i~0asr~uniformlywithrespecttox~R . This implies that lira M i] (x ~ (s)-- xp (s)) (gz~ (s, xp ( s ) ) - - ~ (s, xp (s))) [ ~ f l V ,
r,p--~oo
0
= O.
Similarly, lira r,p~
M ~ II~" (s, xP (s))--~P (s, xp (s))ll2,~dV~
=0.
0
lira M i [[~r (s, xp (s))--~p (S, .Y~P(S)), ~r (S, X r (S)) __~r (S, X p ( S ) ) ] l , s a V s ~- O.
It i s now c l e a r t h a t (3.15) f o l l o w s f r o m (3.16). quence r(i) sueh that for all Markov times T
F r o m (3.15), in t u r n , we o b t a i n the e x i s t e n c e of a s u b s e -
N I x ""+" (~) - x "l'~ (~)12,~ < 2 -~. F i n a l l y , a s in L e m m a 3.4, f r o m t h i s we o b t a i n the u n i f o r m e o n v e r g e n c e of x r ( i ) ( t ) with r e s p e c t to t on any f i n i t e t i m e i n t e r v a l , and b y p a s s i n g to the l i m i t in (3.11) a l o n g t h i s s u b s e q u e n c e r ( i ) we c o m p l e t e the p r o o f of the [ e m m a . P r o o f of T h e o r e m 3.1. The f u n c t i o n s fi, 1] c o n s t r u c t e d in L e m m a 3.3 f o r g i v e n n > 0 we d e n o t e b y a n, b n. By the p r e c e d i n g [ e m m a f o r any n t h e r e e x i s t s a s o l u t i o n of the e q u a t i o n
1259
l
t
x ~ (t) = x o + l an (s, x ~ (s)) d A (s) + S lan (s, x n (s)) dlT~ (S). 0
0
Let ,n=inf{t:lxn(t)[>n}, Zn'm=TnAZ'~. By v i r t u e of the f a c t that a~(t, x ) = a ( t , ]xl -< n , the p r o c e s s e s x ~ (tA*n'm), x m ( t A z ~'m) s a t i s f y the s a m e e q u a t i o n
x), On(t, x ) = b ( t , x) f o r
dz (t) = Zt<~n.,na (t, z (t)) d A (t) + Z,<~n,mb (t, Z (t)) dm (t). H e n c e t h e s e p r o c e s s e s c o i n c i d e , and xn(t) = x m ( t ) on {t -< r n , m} (a. s . ) . T h e r e f o r e , r n -< r m (a. s.) f o r n -< m , t h e r e e x i s t s ~ = l i m * ~ ( a . s . ) , f o r a l m o s t a l l w f o r t < r the p r o c e s s x ( t ) = l i m x n ( t ) is d e f i n e d , and f o r n -+oo
n - - + co
a n y n and a l l t (a. s.) t, A x n
x(tA*n)=xn(tAv~)=x~
t A *n
I a(s,x(s))dA(s)-[-
I b(s,x(s))dm(s).
0
0
It r e m a i n s to p r o v e t h a t r = oo. We s e t
A s we h a v e done r e p e a t e d l y , b y m e a n s of (3.4) f r o m I t o ' s f o r m u l a we find that o~
M ( x ~ (v)) n 2xp,nZ~<~-.< MX~% @ M f Kt~tdVt < o0. 0
F r o m t h i s we find
n2Mq~nZn< ~
(a.s.)
a s n - - ~o. The p r o o f of the t h e o r e m i s c o m p l e t e . R e m a r k 3.1. The a s s u m p t i o n of the c o m p l e t e m e a s u r a b i l i t y of a f t , x), b ( t , x), Kt(R) w a s i m p o s e d o n l y f o r u n i f o r m i t y and in o r d e r t h a t the c o r r e s p o n d i n g s t o c h a s t i c i n t e g r a l s with r e s p e c t to din(t) be d e f i n e d and t h a t the i n t e g r a l s w i t h r e s p e c t to dA(t) and dVt be $ ' , - c o n s i s t e n t . It is e a s i l y s e e n , e . g . , t h a t T h e o r e m 3.1 h o l d s when d a f t ) << dr, a ( t , x) i s m e a s u r a b l e in (t, co) and l i g ' t - c o n s i s t e n t a s a r e t h o s e e l e m e n t s of the m a t r i x b which a r e m u l t i p l i e d b y d m i ( t ) with d <m i }t << dt, w h i l e the r e m a i n i n g e l e m e n t s of b a r e c o m p l e t e l y m e a s u r a b l e , and Kt(R) is p r o g r e s s i v e l y m e a s u r a b l e . 4.
Uniqueness
Theorem.
Finite-Dimensional
A Priori
Estimates.
Approximations
In t h i s s e c t i o n we p r o v e T h e o r e m s 2.2 and 2.3 and p r e p a r e f o r the p r o o f of T h e o r e m 2.1. P r o o f of T h e o r e m 2.2.
By T h e o r e m 1.3.2 f o r yn = u n _ u 0 we find f o r a l l t (a. s.)
!
t gn (t)l ~ = S {2 (v n (s) -- v 0 (s)) [A (v n (s), s)-- A (e 0 (s), s)] + [B (v n (s), s) -- B (v 0 (s), s)[~} ds + m 7 + ] u~ -- ug 15, 0
w h e r e m~ is a l o c a l m a r t i n g a l e
with m n = 0.
We u s e t h i s e q u a l i t y and a p p l y I t o ' s f o r m u l a to c o m p u t e
lyn(t)I~te - K t . A f t e r t h i s , we use the m o n o t o n i c i t y of (A, 13) [condition (A2)] and the e q u a l i t y y = v n - v ~ [ a . e . (t, co). Then f o r s o m e l o c a l m a r t i n g a l e s m~ y we o b t a i n lyn(t)P~i e - K t -< lu~ - u01 0 g2 + m~ ~ -= ~n(t) (a. s . ) . We s e e t h a t the l o c a l m a r t i n g a l e s ~n(t) a r e n o n n e g a t i v e . a s i s known, f o r any e > 0
H e n c e (n(t) is a s u p e r m a r t i n g a l e ,
w h i l e , s i n c e M{n(0) ~ 0,
lira {sup M~ n (t) + P {sup ~n (t) > ~}} = 0. n~
t~T
t~T
A p p l i c a t i o n o f the i n e q u a l i t y lyn(t)I~t -< ~n(t)eKt c o m p l e t e s the p r o o f of T h e o r e m 2.2. P r o o f of T h e o r e m 2.3. L e t v be s o m e s o l u t i o n of Eq. (2.1) and let u be i t s c o n t i n u o u s m o d i f i c a t i o n in H. By T h e o r e m 1.3.2 f o r a l l t E [0, T] on a s i n g l e s e t of full p r o b a b i l i t y
S
l u (t)[~, = l uo I~, +2 i ~ (s) A (~ (s)) ds + 2 ' ~ (s) (B (~ (s)) dw (s) + dz (s)) + ( S B (v (s)) dw (s) + z ) ,. 0
1260
0
0
(4.1)
Let
-~. = inf {t: 1u (015 > n} AT. It i s o b v i o u s t h a t t h e s t o c h a s t i c i n t e g r a [ on the r i g h t s i d e of (4.1) i s a l o c a l m a r t i n g a l e with r e s p e c t to the s e q u e n e e Tn. T h e r e f o r e , tar n
tar n
A41u it A,~)I5 = M iuo I~,+ 2M ~ v (s) A iv (s)) ds + M .fib (v (slG ds + m lz (t A*~)I5 0
0
+2M(ti'nB(v(s))dw(s),
z(tAvn))t 1.
H e n c e , by a s s u m p t i o n (A 3) IA~" n
M I u ( t A ~-)15 ~< M [uo !5-- a M ~ I v (s)i~ds 0
tA*~ (tA*n ) + M [ (i (~) + set. (~)15) <s~+ M I~/t n..)15 + 2M 1 ! B i~, (~)) aw i~), ~ (t n~.) '0
(4.2) H ~
We now m a k e u s e of the e l e m e n t a r y i n e q u a l i t y ~ T1 b2 (~ > 0). 2ab 4 s ~~ 2~-
(4.37
A p p l y i n g it to the l a s t t e r m in (4.2), we o b t a i n t A'r n
0 t A~Cez
t A'r n
0
(4.4)
0
On the o t h e r h a n d , in v i e w of i n e q u a l i t y (2.4) t A'~ n
M
tAX n
(4.5)
f IB(v(s))l~ds~
C o m b i n i n g (4.4), (4.5), and c h o o s i n g e s u f f i c i e n t l y s m a l l , f o r s o m e a l > 0 we o b t a i n MIu(tA,~)
kh~<M]u0[~--~M [
lv(s)l~ds+ c
M
o
f(s)ds+M
{ z } r@ f b
Mlu(sAv~)lhds .
(4.6)
F r o m t h i s by the G r o n w a l l - B e l l m a n l e m m a
sup M ]u (t A T~) ]~ ~
f(s) d s + M ( z ) r 0
9
(4.7)
,
T h u s , f r o m (4.6) a n d (4.7) it f o l l o w s t h a t
A.
(
)
2 M i f (s)ds+ M ( z ) r 9 supMItt(t/k%)l~+ M S [v(s)]~ds-%c M luoi.+ I ~.~T
0
A s n ~ oo, "rn t e n d s to T, b e c a u s e o f t h e c o n t i n u i t y of ]u(t) [ H 2 int. Therefore, in (4.8) (by F a t o u ' s [ e m m a and the m o n o t o n e c o n v e r g e n c e t h e o r e m ) , we o b t a i n
sup Mlu(t)lh+ M S lv(s)l~ds4c tKT
0
(4.8)
0
p a s s i n g to the l i m i t a s n ~
M iUol5-}- M S f (s) ds+ M K Z ) r 0
.
(4.9)
j
To c o m p l e t e the p r o o f of T h e o r e m 2.3 it r e m a i n s to show t h a t an a n a l o g o u s e s t i m a t e a l s o h o l d s f o r M sup ] u (t) ]5, F o r t h i s we note t h a t f r o m (4.1) b e c a u s e of (A 3) and (4.3) it f o l l o w s t h a t t<:T
1261
M sup lu (t A ~ ) I ~ < M 1Uo I~+ M j" ( f (s) + K ]tt (s) 12H)ds t~T
+2Msup t~,T
li
0
~(s)(B(v(s))
-]-2M
;
B(v(s))dw(s)
0
0
(4,10)
+2M(z}r.
U s i n g the B u r k h o i d e r i n e q u a l i t y ( T h e o r e m 1,2.5) and (4.3), we o b t a i n Msup I~T
L
S Zt(s)(B(v(s))
..<334
0
(?
[u(s)[~d<
;
B(v(r))dw(r)+z)~
0
~<w ~M sup l u (t A ,c~)D + gge M
N
U
T
3
3
)
B (v (s)) dw (s) + z (T)
t~T
..< ~ ~M sup [ u (t A ~,) l-2 + T M f [ B (v (s)) Z
T
t~T
z
(4.11)
0
Combining (4.10) and (4.77) and choosing ~ sufficiently small, we obtain on using (4.5), (4.9) Msupjtt(tAT~)[~'4c
f(s) d s + M ( z ) r
Mlttol~+M
t,,~ T
9
0
B e c a u s e of the c o n t i n u i t y of I u(t)lH, the a s s e r t i o n of T h e o r e m 2.3 f o l l o w s f r o m this on the b a s i s of F a t o u ' s l e m m a . The t h e o r e m i s p r o v e d , We proceed finite-dimensional
to prepare spaces.
for the proof of Theorem
2.1.
For
this we approximate
Eq. (2.1) by equations
in
We choose and fix orthonorma[ bases {hi}, {el} in the spaces H and E, respectively. We here assume h i e V, i = I, 2 .... , while ei are the eigenvectors of the covariance operator Q of the Wiener process (this is possible, since Q is a completely continuous operator). Suppose that e i correspond to the eigenvalues k i -> 0 of the operator Q. It is obvious that if h i > 0, then X~I/2(w(t), ei) E is a one-dimensional (standard) Wiener process, and for different i these processes are independent. For n -> 1 we consider
the following
u~(t)=(hzuo)H+ I thA
system
tts
of stochastic
s as+
0
equations
hi, B '~
for (Uln(t) ..... an(t)) e Rn:
tt~(s)hk, s
0
ej
d(w(s), ej)e /H
+ ( l h, z (t))u, i~- 1. . . . . n, tC[0, T]. L E M M A 4.1.
S y s t e m (4.12) h a s a unique ~r't - m e a s u r a b l e
(4.12)
s o l u t i o n which is c o n t i n u o u s in t.
P r o o f . [t s u f f i c e s to show t h a t s y s t e m (4.12) can be w r i t t e n in the f o r m (3.l) and to v e r i f y that the a s s u m p t i o n s u n d e r which T h e o r e m 3.1 w a s p r o v e d a r e s a t i s f i e d . We note i m m e d i a t e l y that the m e a s u r a b i l i t y conditions are verified by using the remark at the end of See. 3, and we leave this verification to the reader. We
choose
d = n, d I = 2n, and we
set a (t, x ) ~- (a' (t, x ) ,
b (t, x ) = (b ~j (t, x ) ,
i = 1 .....
i ~ 1 .....
n),
n, j = 1 . . . . .
2n),
m (t) ~ (m f (t), i = 1 . . . . . 2n), A (t) = t, nz ~(t) = (w (t), ez)e, rn ~+~(t) = (z (t), th)u, 1 < i < n,
a ~(t, x) = thA
xHtj, t , 1 ~ i ~ rt, \]~i
b~J(x)~- 6n+t. ] , 1 4 i 4 n ,
n+l~j..<2n,
w h e r e 6 k , j is the K r o n e c k e r s y m b o l . With t h i s n o t a t i o n it is o b v i o u s t h a t (4.12) can be w r i t t e n in f o r m (3.1). A s in S e c . 3, we i n t r o d u c e the q u a n t i t i e s ciJ(t), c~(t, x), B(t, x) f o r V t = t + ( w } t + ( z } t . It is c l e a r t h a t (3.2) f o l l o w s f r o m (A 4) and (A~). Further,
1262
we c h o o s e x , y ~ R n a n d s e t u
~xthi,
v~'~gqh.
It is not h a r d to s e e that
[I~ (t, x ) - ~ (t, y)II~ dr, =- 2
(h,, IB (u, t)
i ,j=l r
-- B (~, 0] ej)~Xjdt 4 ~ X] lIB (u, t) -- B (~, t)] e] lSdt = ]B (u, t)-- B (v, t)!2o dt.
(4.13)
j=l
H e n c e , by condition (A2)
{2 ( x - - g) (a (t, x ) - - a (t, g)) + II~ (t, x) -- IB(t, g)If2} dVt -.< 2 (u -- ~) [A (u, t) -- A (~, t)l dt + [ B (u, t) -- B (v, t)]~dt < K tu-- ~ [2Hdt-~ K ( x -- y)2 dr. T h e r e f o r e , condition (3.3) is s a t i s f i e d . M o r e o v e r , the continuity of a(t, x) in x can be d e r i v e d f r o m L e m m a 2.1, and the continuity of b(t, x) f r o m the continuity of B(u, t) which, in t u r n , follows f r o m (A2). It r e m a i n s to c h e e k condition (3.4). We set l
(t) = ~ b (s, x)drn (s). 0
F o r any ~ > 0 we have by (2.4) in a n a l o g y to (4.13) d ([)t..<(t+s
;.j(h. B(u, t ) e ) ~ d t + ( 1
)
+T) Zd
t,]=I
+(2 w h e r e c does not depend on e.
( m'*+i ) t~
i=I
+J-)d< z >
(/(t)+ f
(4.14)
We note f u r t h e r that (2x{z (t, x) + l] iB (t, x) ]l2) dV t = 2uA (u, t) dt 2 d ( ~ ) t.
(4.1 5)
Combining (4.14), (4.15) with condition (A 3) and then c h o o s i n g e sufficiently s m a r t , we conclude that
(2xa (t, x ) + l i ~ (t, x) [i~)dVt ~
hn}, by ~rn the p r o j e c t i o n o p e r a t o r of E onto
n
{e, . . . . .
en} , and we set un(t) = ~,~ain(t) h , . I r i s then obvious that (4.12) is e q u i v a l e n t to the equation i=l t
Un(t) = ao + .i HnA (an (s), s) ds -J- 5 HnB (an (s), s) & d w (s) + llnz (t). 0
(4.16)
0
We r e q u i r e the following a s s e r t i o n . T H E O R E M 4.1.
1) T h e r e e x i s t s a c o n s t a n t e such that f o r all n s i m u l t a n e o u s l y Msup!a,(t)l~+Ms
dt
t..< T
0
Mluol~+Mj'f(t)dt+M 0
( z)r
;
t
2)
Me-.tlu~(t)i~=Min,~Uo!~+MSe-..{2a.(s)A(u,,(s))V_lH.B(a,~(s))~.l~_c[u.(s)l~}ds 0
+ 2 N ~o
(r))
(r) ) ~+
0
e-~d ( H~z ) , 0
.for all t _< T and f o r any c o n s t a n t c. The p r o o f of this t h e o r e m is a l m o s t a l i t e r a l repetition of the p r o o f of T h e o r e m 2.3, and we leave it to the r e a d e r . We point out only that
a,,I1.A (an)=u.A (a.), In.B (a.) ~./Q--
1263
5.
Passage
Existence
to of
the
Limit
a Solution.
by
the
Monotonieity
Markov
Method.
Property
In t h i s s e c t i o n we c o m p l e t e the p r o o f of T h e o r e m 2.1 and p r o v e T h e o r e m 2.4. We denote by 3- the ~ - a i g e b r a of p r o g r e s s i v e l y m e a s u r a b l e s e t s on [0, T] x a and by W i t s c o m p l e t i o n w i t h r e s p e c t to the m e a s u r e L x P (L is L e b e s g u e m e a s u r e on [0, T]). We s e t S = ([0, T] • o~, L x P ) . F r o m t h e r e s u l t s of the p r e c e d i n g s e c t i o n ( T h e o r e m 4.1) it f o l l o w s t h a t t h e r e e x i s t s a s u b s e q u e n e e {v} a l o n g which f o r s o m e u, v, u~ (T) uv-+u weakly in L 2 (S;/-]), uv-->r weakly in Lp(S; V),
(5.1) (5.2)
Uv (T) -+tt~ (7") weakly in L2 (Q, J'r, P ; / u
(5.3)
By T h e o r e m 4.1, (A3), a n d (A 4) it m a y b e a s s u m e d t h a t A (uv) -+ A= weakly in Lq (S; V*), llvB (uv) ~v-+B~ weakly in L2 (S; LQ (E; H)).
(5.4) (5.5)
G e n e r a l l y s p e a k i n g , u, v, u ~ ( T ) , A ~ , B ~ a r e c l a s s e s of f u n c t i o n s , b u t in the c l a s s e s u, v, A ~ , B ~ it is p o s s i b l e to c h o o s e a p r o g r e s s i v e m e a s u r a b l e r e p r e s e n t e r and in the c l a s s u~(T~ a 5 r r - m e a s u r a b l e r e p r e s e n t e r . We h e n c e f o r t h c o n s i d e r j u s t t h e s e r e p r e s e n t e r s and p r e s e r v e f o r t h e m the n o t a t i o n of the c o r r e s p o n d i n g c l a s s e s . We note a l s o t h a t s i n c e the i m b e d d i n g of V in H is d e n s e , it f o l l o w s t h a t u = v f o r a . e . (t, co). F u r t h e r , it is w e l l known t h a t any s t r o n g l y c o n t i n u o u s , l i n e a r o p e r a t o r is w e a k l y c o n t i n u o u s . f o r e , b e c a u s e of (5.5) t
There-
t
S U~B(u~ (s)) ~,dw (s)-~ ~ B~ (s) aw (s) 0
(5.6)
0
w e a k l y in L2(S; H) and in L2(~, ~v't,P; H ) f o r e a c h t. L e t y b e any b o u n d e d r a n d o m v a r i a b l e , and l e t ~b(t) b e a bowaded function on [0, T]. (4.16) t h a t f o r e a c h h i and v -> i
M S v,(t)(ttv(t), h~).dt=M 0
y~2(t)h~ uo+ 0
j'A(u,(s))ds
+
0
It f o l l o w s f r o m
II.B(uv(s))r~vdw(s)+z(t)
dt.
0
P a s s i n g to the l i m i t a c c o r d i n g to (5.2), (5.4)-(5.6) in t h i s e q u a l i t y , we o b t a i n
MS y~(t)(v (t), h~)Hdt= 214o;Y*h~ Uo+
A~o(s)ds +
0
B~(s) dw(s)+z(t)
at.
0
F r o m t h i s it f o l l o w s that f o r a . e . (t, co) t
t
v (t) = Uo+ ~ A~ (s) as + ~ B~ (s) dw (s) + z (t). 0
(5.7)
0
In the s a m e w a y , u s i n g (5.3)-(5.6), we find t h a t ( a . s . ) T
T
u~ (T) = uo + ~ A~ (s) ds + S B~ (s) dw (s) + z (T). 0
(5.8)
0
By T h e o r e m 1.3.2 t h e r e e x i s t s an 5 r r - c o n s i s t e n t function c o n t i n u o u s in t w i t h v a l u e s in H w h i c h c o i n c i d e s with v(t) f o r a . e . (t, w) and i s e q u a l to the r i g h t s i d e of (5.7) f o r a l l t e [0, T] and co ~ f~' [P(f2') = 1]. We i d e n tify it with u; t h i s is p o s s i b l e , s i n c e u v [ a . e . (t, co)]. In view of (5.8), we t h e n have
.~ (r)=.(r)
(a. ~.).
(5.9)
By t h i s s a m e T h e o r e m 1.3.2 f o r a l l (t, co) ~ [0, T] x Q, l
!
I u (t)]~ = 2 S ~ (s) A~ (s) as + 2 S ~ (s) (B~ (s) dw (s) 4- dz (s)) + ( i' B~ (s) dw (s) + z }t @ IUo15. 0
0
(5.10)
0
The r e s t of the a r g u m e n t s a r e t y p i c a l f o r the t h e o r y of m o n o t o n e o p e r a t o r s . T h e y do n o t r e q u i r e the s p e c i a l a t t e n t i o n of the r e a d e r , s i n c e on a f i r s t r e a d i n g they p r o d u c e the i m p r e s s i o n of a c o l l e c t i o n of b a n a l t e c h n i q u e s m y s t e r i o u s l y l e a d i n g to t h e r e q u i r e d r e s u l t .
1264
L e t y(t, co) be any S t ' v - c o n s i s t e n t function m e a s u r a b l e
in (t, co) with v a t u e s in V w h e r e b y
T
"< S (I v (t)l~ + t y (t)l~,) at < oo.
(5.11)
0
We s e t T
O~ = Ill ~"e -c~ {2 (u~ (t) -- g (t)) (A (tl~ (t))-- A (V (t)) -- c ] u~ (t)-- g (t)]~ + I II~B (z~ (t)) a*~-- II~B (V (t)) :*~ l~} dt. 0
F r o m 0k 2) it follows that f o r a s u i t a b l e choice of c
O,,<0. Further,
(5.12)
we r e p r e s e n t O~ in the f o r m O v = O ~ + O ~ w h e r e T
O~ = M S e-~ {2u~ (t) A (uv ( t ) ) - c [uv (t)15 + I llvB (u, (t)) n,. J~} dt, 0 and 9
is defined by the d i f f e r e n c e .
By T h e o r e m 4.1 T
7"
OI = Me-cr j u~(T)tn--M '2 'I II,Uo4n 2 _ 2 M S e - , d < S II,B(u, (s)) ~,dw (s), z > t --Mje-~176 0
We r e m a r k that
0
o
7"
o
=Me -~
;"
IIvB(ttv(s)).%dw(s), z(T) H47 cMo
II,B(~t,(s)).%dw(s), z(t) ttdt.
F r o m this and (5.6) it follows that T
T
f whereS:limMlu.(T)[~--Mlu(T)[~>O
< >
>
by (5.3).
On the o t h e r hand, b e c a u s e of (5.10), T
/4e-~r I u (T) I~,-- M i u01 ~ = M I e-~t {2v (t) A~ (t) -- c I ,, (012 + IB ~ (t)[ ~} a t 0 T
T
0
0
0
C o m p a r i n g (5.13) and (5.14), we see that T
l imO~ = M f e-~' {2~ (t) A~ (t) -- c in (t) t 5 +tBoo (t) I ~} dt + ~e-or.
(5.15)
0
F u r t h e r , f r o m (5.1), (5.2), (5A), and (5.6) it follows that T
l~m o~= ~4 f e-~' Pv (0 A (V(0)-- 2V(0 A~ (0--2~ (0 A (V(0) 0
+ 2o (u (0, ~ ( t ) ) ~ - c I v (0 I~ - 2 (B~ (0, B (v (0))Q + t B (~ (t)) 1~] at. Combining (5.15) and (5.16), we find, in view of (5.12),
(5.161
T
M f e-~'{2(v(t)--g(t))(A~(t)--A(g(t)l)-clu(t)--g(t)i~+!B=(t)--B(g(t))l~}
dt-[-;e-Cr40.
(5.17)
0
Setting y = v in (5.17), we see that B~ (t) = B(v(t)) [ a . e . (t, w)], and
8:Iim M[av(T)[~--MI~(T)I~=O.
(5.1s)
1265
On the o t h e r h a n d , it f o l l o w s f r o m (5.17) t h a t T
M I e-c[ {2 (v (t) - - g (t)) (Am ( t ) - - A (g (t))) - - c t u ( t ) - - g (t) i5} dt ..< O.
(5.19)
0
S u p p o s e now t h a t x ( t , co) is any p r o c e s s with v a l u e s in V w h i c h s a t i s f i e s an i n e q u a l i t y a n a l o g o u s to (5.11), a n d l e t y = v - Xx, ? , ~ R + . F r o m (5.19) we then o b t a i n the i n e q u a l i t y T
M i e-c[ {x (t) (A~o(t) - - A (v (t) +)~x (t))) - - c)~ l x (t)
dt ..< O.
0
L e t t i n g X go to z e r o , we find f r o m t h i s by (A 4) and the L e b e s g u e t h e o r e m t h a t T
A[ I x (t) (A~o(t) - - A (v (t))) dt ..< 0. 0
Since x i s a r b i t r a r y , f r o m t h i s it f o l l o w s t h a t A~ (t) = A (v(t)) [a. e. (t, w)] w h i c h t o g e t h e r with (5.7) and the f a c t p r o v e d e a r l i e r t h a t Boo (t) B(v(t)) [a. e. (t, w)] c o m p l e t e s the p r o o f of T h e o r e m 2.1. H.
We n o t e , m o r e o v e r , t h a t we have p r o v e d e q u a l i t y (5:18) in w h i c h u is a c o n t i n u o u s m o d i f i c a t i o n of v in On the o t h e r h a n d , f r o m (5.3) we have Mltt(T)12H
lim M ] uv (T)[5 = M [ tt (T)i~,. u ~,-co
R e t u r n i n g a g a i n to (5.3) and r e c a l l i n g t h a t a w e a k l y c o n v e r g e n t s e q u e n c e c o n v e r g e s if the n o r m of the l i m i t is e q u a l to the l i m i t of the n o r m , we c o n c l u d e that lira M i "~ ( r ) - - ~ (r)[,~ = 0. a;_+~
H e r e u(T) d o e s n o t d e p e n d on the s e q u e n c e {v} b y T h e o r e m 2.3. We have t h u s p r o v e d that any s u b s e q u e n c e of the s e q u e n c e { u n ( T ) : n = 1, 2, 3 . . . . } c o n t a i n s a s u b s e q u e n c e c o n v e r g i n g s t r o n g l y in L2(O, ~r'r, P; H) to u(T). T h i s i m p l i e s t h a t a l o n g the o r i g i n a l s e q u e n c e un(T) ~ u(T) a s n ~ oo. Since in p l a c e of s e g m e n t [0, T] we c o u l d c o n s i d e r any s m a l l e r s e g m e n t , we have p r o v e d the f o l l o w i n g r e s u l t . C O R O L L A R Y 5.1. s o l u t i o n s of E q s . (4.16).
L e t v b e a s o l u t i o n of (2.1), let u be i t s c o n t i n u o u s m o d i f i c a t i o n in H, and l e t un be Then f o r any t ~ T lira M I tt~ (t) - - it (t)[~ = 0.
P r o o f of T h e o r e m 2.4.
T o g e t h e r with Eq. (2.1), we c o n s i d e r the e q u a t i o n t
t
v (t) = tt~ @ f A (v (r), r) dr -~- I B (v (r), F) dw (r), 8
(5.20)
$
w h i c h we s o l v e f o r t ~ [s, T] and M l u s l h < oo. The d e f i n i t i o n s of a V - s o l u t i o n and H - s o l u t i o n a r e m a d e f o r Eq. (5.20) in the u s u a l way. A l l the a s s e r t i o n s p r o v e d f o r Eq. (2.1) c a r r y o v e r to Eq. (5.20) in a n a t u r a l w a y . In p a r t i c u l a r , it h a s an H - s o l u t i o n u(t) = u(t, s , Us). F o r u s we c h o o s e a n o n r a n d o m e l e m e n t x ~ H, a n d f o r t ~ [s, T] and b o u n d e d B o r e [ f u n c t i o n s f(h) d e f i n e d on H, we s e t
M , , x f (tt (t)) = M f (tt (t, s, x)).
(5.21)
To p r o v e the t h e o r e m it o b v i o u s l y s u f f i c e s to show t h a t (5.21) d e f i n e s a B o r e [ function of x, and f o r t h e H - s o l u t i o n u(t) of Eq. (2.1) M { f (it (t))j ~ } = M~,u(~)f (it (t)) (a. s.).
(5.22)
The r e q u i r e d p r o p e r t y of m e a s u r a b i l i t y is e a s i l y p r o v e d b y m e a n s of a c o r r e s p o n d i n g a n a l o g of T h e o r e m 2.2 w h i c h s h o w s that f o r c o n t i n u o u s f the function M s , x f ( u ( t ) ) is a l s o c o n t i n u o u s in x. [t s u f f i c e s to p r o v e the r e l a t i o n (5.22) o n l y f o r c o n t i n u o u s f. M o r e o v e r , f o r s i m p l i c i t y we a s s u m e t h a t s - 0. T h i s can a l w a y s be a c h i e v e d by c h a n g i n g the t i m e v a r i a b l e .
1266
"We n o t e , f i r s t o f a l l , t h a t u(t, 0, x) d o e s not d e p e n d on 5Z-o. T h i s f o l l o w s f r o m C o r o l l a r y 5.1 sand the f a c t t h a t in S e e . 3 s o l u t i o n s of s t o c h a s t i c e q u a t i o n s w e r e c o n s t r u c t e d in f i n a l a n a l y s i s b y m e a n s of p a s s i n g to the l i m i t f r o m e q u a t i o n s w i t h c o e f f i c i e n t s t h a t s a t i s f y a L i p s c h i t z c o n d i t i o n f o r w h i c h the i n d e p e n d e n c e of a s o l u t i o n with n o n r a n d o m i n i t i a l d a t a f r o m 5to is known ( s e e , e . g . , [30, 16]). F u r t h e r , we a p p r o x i m a t e u0(co) b y s t e p f u n c t i o n s u~(co) and l e t Fn be the s e t of v a l u e s of u~(c0). It is e a s y to s e e t h a t u(t, 0, u n) and
X~P n
satisfy the same equation and therefore coincide (a. s.). F r o m this and the fact that u(t, O, x) is independent of ~0, ft follows that
M {f (~ (t, 0, ~))1Y0} =.% ~,,/(u (t)) (~. ~.). '
0
L e t t i n g h e r e n ~ ~ and u s i n g T h e o r e m 2.2, we o b t a i n (5.22) f o r c o n t i n u o u s f. is complete. CHAPTER ITO 1.
STOCHASTIC
PARTIAL
The p r o o f of the t h e o r e m
III
DIITFERENT[AL
EQUATIONS
Introduction
T h i s c h a p t e r is d e v o t e d to a p p l i c a t i o n s of the r e s u l t s of Chap. lI to s t o c h a s t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s . We c o n s i d e r the f i r s t b o u n d a r y - v a l u e p r o b l e m f o r n o n l i n e a r e q u a t i o n s of a r b i t r a r y o r d e r and the Cauchy p r o b l e m f o r l i n e a r e q u a t i o n s of s e c o n d o r d e r . The l a t t e r m e r i t s p e c i a l a t t e n t i o n , s i n c e s u c h an i m p o r t a n t p r o b a b i l i s t i c p r o b l e m a s the p r o b l e m of f i l t r a t i o n f o r d i f f u s i o n p r o c e s s e s r e d u c e s to the i n v e s t i g a t i o n of e q u a t i o n s of t h i s t y p e ( s e e , e . g . , [31, 27, 85])9 The r e s u l t s p e r t a i n i n g to n o n l i n e a r e q u a t i o n s a r e r e l a t e d to [12] w h e r e e q u a t i o n s without s t o c h a s t i c t e r m s a r e c o n s i d e r e d , w h i l e e l e m e n t s of the t h e o r y of l i n e a r e q u a t i o n s a r e e x p o u n d e d a c c o r d i n g to o u r w o r k [261. At[ c o n s i d e r a t i o n s a r e c a r r i e d out in S o b o l e v s p a c e s 9 f r o m the t h e o r y of t h e s e s p a c e s .
We r e c a l l the d e f i n i t i o n s and s o m e b a s i c f a c t s
L e t R d b e d - d i m e n s i o n a l E u c l i d e a n s p a c e with a f i x e d b a s i s . We denote b y a , / 3 , T, c~i, Bi, Ti (i = 1, 2, 9 ) c o o r d i n a t e v e c t o r s in t h i s s p a c e and a l s o the nuI[ v e c t o r . If a is the nul! v e c t o r , then D ~ i s the i d e n t i t y o p e r a t o r , while if a is the i - t h b a s i s v e c t o r , then D a = 8/Dx i. S u p p o s e , f u r t h e r , t h a t G is a d o m a i n in R d, F is the b o u n d a r y of G, and m i s an i n t e g e r , m ~ 1, p ~ (1, ~ ) . D e f i n i t i o n 1.1.
The S o b o l e v s p a c e w~n(G) i s the s p a c e of r e a l f u n c t i o n s d e f i n e d on G with finite
noryn
I'"
~ff*~' 9
r/1
]
w h e r e D c~l . . . D c~m a r e g e n e r a l i z e d d e r i v a t i v e s , and
Ilgl[p-
Ig(x)l pclx
0m
D e f i n i t i o n 1.2. The S o b o t e v s p a c e Wp (G) i s the c l o s u r e of C o (G) (the s p a c e of i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s with c o m p a c t s u p p o r t s in G) in the n o r m ll-[lm, p. T H E O R E M 1.1. n o r m II~llm,p.
0m
The s p a c e s Wp (G), w~n(G) a r e s e p a r a b l e ,
r e f l e x i v e B a n a c h s p a c e s r e l a t i v e to the
T H E O R E M 1.2 (Sobolev I m b e d d i n g T h e o r e m ) . If the d o m a i n G i s b o u n d e d , i t s b o u n d a r y F is r e g u l a r , and 2(d - mp) -< p d , then W / , ( G ) c L 2 ( G ) , while t h i s i m b e d d i n g i s d e n s e and c o n t i n u o u s . In p a r t i c u l a r , t h e r e e x i s t s a c o n s t a n t e s u c h t h a t Ilult 2 -< e l l u l l m , p f o r a l l u E W~n(G). P r o o f s of t h e s e t h e o r e m s can b e found in [38, 34, 70]; the c o n d i t i o n s on the b o u n d a r y w h i c h we have c a l l e d r e g u l a r a r e a l s o p r e s e n t e d t h e r e . The f o l l o w i n g a s s e r t i o n is o b v i o u s .
1267
T H E O R E M 1,3. in L 2(Rd).
The Sobolev space w~n(R d) is a H i [ b e r t s p a c e ;
it is continuously and d e n s e l y imbedded
We f u r t h e r need below the s o - c a l l e d F r i e d e r i e h s inequality (see, e . g . , [38, 7 0 ] ) w h i c h is given in the following t h e o r e m . T H E O R E M 1.4. Suppose that the domain G is bounded, and its b o u n d a r y is r e g u l a r . 0m a c o n s t a n t c > 0 such that f o r any u E Wp (G)
IlUllm.,-.
~
Then t h e r e e x i s t s
]ID~'...D ul],.
Icql4-... +lC~m[=m
The next fact is also well known; it follows e a s i l y by m e a n s of F o u r i e r t r a n s f o r m and the P a r s e v a l equality. T H E O R E M 1.5.
T h e r e e x i s t s a c o n s t a n t c > 0 such that f o r any u E Wm(R d)
1]ztN"~'2
The negative n o r m of an e l e m e n t f 6 Lq(G) is
IIf IJ_~,~ = sup (f, u)0, w h e r e the s u p r e m u m is taken o v e r the set of all functions u e @~n(G) f o r which Ilullm,p = 1, and
( f , U)o= f f (x) ,t (x) dx. G
Definition 1.4.
The space with n e g a t i v e n o r m w[tm(G) is the c o m p l e t i o n of Lq(G) in the n o r m II'l]-m,q. 0
It is c l e a r that f o r f ~ Lq(G), u E wn~(G) we have (f, u) 0 -< []fllq'lfUllp - [lfllq'llullm,p. Deftnitinn 1.4 is t h e r e f o r e c o r r e c t . F r o m the definitions given it follows i m m e d i a t e l y that t h e r e a r e the n a t u r a l imbeddings 0
Wp (G)~Lplu)~ p ((7). 0m
01i3_
The duality b e t w e e n Wp (G), W~pre(G) is defined by m e a n s of the s c a l a r p r o d u c t in L2(G): if v E W p (G), m* - 72" II ---~~, (~ then we s e t <> = l i r a (v~, v])0. v*EW~ ~ (G), v~EC~ (G), vkELq (G), IIv ~ - v [l~.p-+0, I[~ ,~,i-~,q n+oo
~ (G) t h e r e is a v * E Wqm(G) such that this It is found that f o r any c o n t i n u o u s , l i n e a r functional on Wp functional is equal to <>, and, c o n v e r s e l y , any c o n t i n u o u s , l i n e a r functional on Wqm(O) e a n b e w r i t t e n a s 0m
<) f o r s o m e v E Wp (G).
0m
M o r e o v e r , f o r v E Wp (G), v* E V~pm(G) ]lVllm,p=sup <>
~* IIw*ll_~,q'
_, v
<<w, v*>> - - s u p , Iiwll~-~,p " _
_
0 m
This m a k e s it p o s s i b l e to identify by m e a n s of << >> the dual space of Wp (G) with Wqm(C).
We note
that u n d e r this identification the duality <( >> p l a y s a d e c i s i v e r o l e . T h u s , f o r p = 2 the space ~r is a H i l b e r t space and, as any H i l b e r t s p a c e , it is identified in the well-known way with its dual space (by m e a n s of its s c a l a r p r o d u c t ) . At the s a m e t i m e , of c o u r s e , wm(G) ;~wTm(G). T h u s , different b i l i n e a r f o r m s m a k e it p o s s i b l e to c o n s t r u c t dual s p a c e s of @m(G) in different w a y s . of T h e o r e m 3.2.
We shall use these c o n s i d e r a t i o n s in the p r o o f
0
,
We identify the dual space of Win(G) with Warn(G), so that the r e s u l t of applying a functional v ~ Wqm(C-) 9 In
~
-L
to an e l e m e n t v ~ Wp (G) we w r i t e <>. This notation d i f f e r s f r o m that used e a r l i e r in Chaps. I and II vv*, b e c a u s e v* m a y be an o r d i n a r y function and then the notation vv* m i g h t be u n d e r s t o o d as the p r o d u c t of the functions v and v*. T H E O R E M 1.6. continuous.
1268
L e t 2(d - mp) -< pal. 0m
If v* E L2(G), v E Wp (G), then
Then
0
W~ (O)~L2 (G)cW~ "~(G), w h e r e
<> = ( v , v*)0.
e a c h imbedding is dense and
2.
First
Stoehastie
Boundary-Value Equations
Problem of
Parabolic
for
Nonlinear
Type
L e t (f~, 5~, P) be a e o m p l e t e p r o b a b i l i t y s p a c e ; let {Yt} be an i n c r e a s i n g f a m i l y of e o m p [ e t e or-algebras i m b e d d e d in 5r; let G be a bounded domain with a r e g u l a r b o u n d a r y o r G = R d, p = 2; 2(d - rap) -< dp, let z(t) be a s q u a r e - i n t e g r a b l e m a r t i n g a l e (relative to g-t) with v a l u e s in L2(G) which is eontinuous in t [in Lz(G)], and let W(t) be a W i e n e r p r o c e s s with v a l u e s in s o m e s e p a r a b l e H i i b e r t space E with e o v a r i a n c e o p e r a t o r Q. By T h e o r e m 1.6 the s p a c e s V - ~ ( G ) , H = L~(G), V* = w q m ( G ) s a t i s f y assLumptions a)-d) of Chap. II. In the c y l i n d e r [0, T] x G f o r fixed T > 0 we c o n s i d e r the p r o b l e m du (t, x, co)= - - ( - - 1)l~'r+'''+l~ml D ~ ' . . . .
. . . . D%~Ac+,...~m( D O ' . . . . . D~mu (t, x, @, t, x, o~) dt + + B ( D ~'" ~ . . ' D ~ t t ( t , x, o)), t, x, o~) dw (t, o~)+dz(t, x, o~)
x~G,
~(0, x, @=[to(X, r
DP~
"Df%-'Uis--O,
for
all
(2.1) (2.2)
~0. . . . . [~-t
(2.3)
such that 11301, -}-.-.@t~-~l~ ~-1, w h e r e S is the l a t e r a l s u r f a c e of the c y l i n d e r ; s u m m a t i o n o v e r all v a l u e s of r e p e a t e d indices ~i is u n d e r s t o o d ; the functions A, B depend on t, x, co, and all d e r i v a t i v e s of u with r e s p e c t to x of o r d e r no g r e a t e r than m ; A and u a r e r e a l f u n c t i o n s ; B is a function with v a l u e s in E; in the s e c o n d t e r m in (2.1) the s c a l a r p r o d u c t in E is u n d e r s t o o d . We a s s u m e t h a t f o r any collection of r e a l n u m b e r s [ - (Stir ..... ~ ) and any e ~ E the functions A(~, t, x, co), B(~, t, x, co)e (B(~, t, x, co), e)E a r e m e a s u r a b l e in (t, x, co), f o r e a c h t ~ [0, T] they a r e m e a s u r a b l e in (x, co) r e l a t i v e to the p r o d u c t of the B o r e [ (r-algebra of R d and S~, and f o r e a c h t, x, co they a r e continuous in . Suppose f u r t h e r that t h e r e e x i s t a c o n s t a n t K > 0 and a nonnegative function f(t, x, co) p o s s e s s i n g the s a m e p r o p e r t i e s of m e a s u r a b i l i t y as A(0, t, x, co) such that f o r all t, x, co, [ , ~i . . . . . C~m 1
I A ...... ~ (~, t, x, ~)1-.< F
(t, x, ~) + K
]~
l ~P...... ~/,-~
(2.4)
P. . . . . . ~ m
]B([, t, x, m) [~ ..< f (t, x, 09 + K
~
lg e...... ' % I ~ + K I [ ~..... 012.
(2.5)
M o r e o v e r , it is a s s u m e d that u0(x , w) is m e a s u r a b l e r e l a t i v e to the p r o d u c t of the B o r e [ a - a l g e b r a of R d and S 0 , and f o r a l l t , w, Uu0112 < ~ , IIf(t)lll < ~ , T
Mll.0iI~<~,
Mj'Hf(t)lhctt< oo. 0
T h e s e a s s u m p t i o n s enable us to give a p r e c i s e m e a n i n g to Eq. (2.1) and the conditions (2.2) and (2.3) in the following m a n n e r . F o r notational c o n v e n i e n c e we o m i t all argnaments of s o m e functions o r only p a r t of t h e m . We note that by (2.4) and the H 6 [ d e r inequality I
(A .......
D",-....
o (N/(t)
II" II;,:/)Jl i1,,.,.
(2.6)
T h e r e f o r e , f o r t ~ [0, T], w ~ _q, u ~ W~n(G) the left side of (2.6) is a l i n e a r functional on W~n(G) and, in 0m
p a r t i c u l a r , on V = Wp (G). all v ~ V
As we knouT, t h e r e e x i s t s a unique e l e m e n t A(u, t, co) ~ V* = Wqm(G) f o r which f o r <
~ - - ( A ...... m(D ~. . . . D~'~tt, t, o~)Dr~'...D~"~V)o .
It is c l e a r that the function A(u, t, co) s a t i s f i e s the m e a s u r a b i l i t y conditions of See. II.2 and conditions (A1), (A 4) of this s e c t i o n . F u r t h e r , we note that f o r g E H b y (2.5) ([ B (D~,.....D%~u, t)[e, g)o~< []f (t)][~12. ]] g tlH + N (Hu ]l~l2 il g ]lu+ IIu I[H[I g ]In).
1269
This i m p l i e s that f o r t ~ [0, T], w ~ ~, u ~ V a c c o r d i n g to the f o r m u l a
(B (u, t, o~)e, g)o~((B (De,.....D~mu, t , . , (o), e)e, g)o, eCE, gCH, B(u, t, oOdZ(E, PI) is defined, and if el, h i a r e o r t h o n o r m a l b a s e s in E , H, r e s p e c t i v e l y , then [[B (it, t)li 2= ~ (B (u, t) e~, h])2~< Ill B (De . . . . . 9D~tt, t)Je II~~ J[f (t)[]~ + N (ll u ll~ + 1]u ]]~).
(2.7)
i]
An a n a l o g o u s inequality o b v i o u s l y holds f o r J113(u, t)Q~/211 2. H e n c e , f o r B(u, t) inequality (11.2.4) holds. M o r e o v e r , it is c l e a r that if u = u(t, co) is a function with values in V which is m e a s u r a b l e in (t, co) and ~ c c o n s i s t e n t , then B(u(t, co), t, co) p o s s e s s e s the s a m e p r o p e r t i e s of m e a s u r a b i l i t y as an e l e m e n t of ~ o ( E , H). T h u s , Eq. (II.2.1) m a y be c o n s i d e r e d f o r the o p e r a t o r s A, B defined above. Definition 2.1. A V - s o l u t i o n (H-solution) of p r o b l e m (2.1)-(2.3) is a V - s o l u t i o n (H-solution) of Eq. (II.2.1). A continuous m o d i f i c a t i o n of p r o b l e m (2.1)-(2.3) in H is defined s i m i l a r l y . R e m a r k 2.1. A function v(t) E V is a V - s o l u t i o n of p r o b l e m (2.1)-(2.3) if and only if it is a p p r o p r i a t e l y m e a s u r a b l e , inequality (II.2.2) is s a t i s f i e d , and f o r any ~ E V t
(v (t), rl)o~(Uo, ~])0--S (A~,. . ~ (D~,. . . D ~ v , s), D ~ . . . D ~ l ) o d s 0 t
-t- ~ ~ B (Dh. . . D ~'v, s, x) ~1(x) d x d w (s) + (z (t), ~1)o 0 Rd
(a.e.: (t, ~)).
(2.S)
In o t h e r w o r d s , a V - s o l u t i o n is a solution of p r o b l e m (2.1)-(2.3) in the sense of the i n t e g r a l identity. o r d e r to o r o v e t h i s , it s u f f i c e s to use R e m a r k (II.2.1) and the fact that f o r , e H, the n o r m 113 (13p l . . . D p m v ( t , x), t, x)IE m u l t i p l i e d by I~/(x)l is integrable o v e r R d, and hence f o r any e ~ E
~B (v, t) e = (~, B (v, t) e). = ~ (B (D~,:...D~",v, t,
x),
e)e~ (x) d x
Rd
~ (~a B (D,, . . . Dgrn v, t, x) ~ (x) d x , e)E. A s i m i l a r r e m a r k holds f o r an H - s o l u t i o n of p r o b l e m (2.1)-(2.3). R e m a r k 2.2. Relation (2.8) is o b t a i n e d if we f o r m a l l y m u l t i p l y (2.1) by ~, i n t e g r a t e o v e r t , x, i n t e g r a t e by p a r t s in x, and use the b o u n d a r y conditions (2.3). In o u r i n t e r p r e t a t i o n of a solution of (2.1)-(2.3) we s t a r t f r o m (2.8) in which t h e r e a r e no b o u n d a r y conditions. They a r e a c c o u n t e d with in (2.8) only t h r o u g h the 0 m e m b e r s h i p of v in the space w m ( G ) . In this connection we note that f o r p > d by one of the Sobolev imbedding t h e o r e m s e a c h function of @m(G~ is equal a l m o s t e v e r y w h e r e to a function which has continuous d e r i v a t i v e s in p t h r o u g h o r d e r m - 1, and these d e r i v a t i v e s a r e a c t u a l l y equal to z e r o on the b o u n d a r y of G. Definition 2.2. We s a y that Eq. (2.1) s a t i s f i e s the condition of s t r o n g p a r a b o l i c i t y if the o p e r a t o r s A, B s a t i s f y conditions (A2), (As) of See. II.2. Conditions which a r e sufficient f o r s t r o n g p a r a b o l i c i t y in t e r m s of the o r i g i n a l functions Ac~ 1... a m (~, t, x, co), 13 (~, t, x, co) will be given below. The next t h e o r e m follows a u t o m a t i c a l l y f r o m the r e s u l t s of See. II.2. T H E O R E M 2.1.
If Eq. (2.1) is s t r o n g l y p a r a b o l i c , then the a s s e r t i o n s of T h e o r e m s II.2.1-II.2.4 hold.
V e r i f i c a t i o n of the condition of s t r o n g p a r a b o l i c i t y in the g e n e r a l c a s e is a p r o b l e m of c o l o s s a l difficulty even f o r B = 0. H o w e v e r , by g e n e r a l i z i n g [12], in c e r t a i n c a s e s it is p o s s i b l e to give simple sufficient c o n d i tions f o r the s t r o n g p a r a b o l i e i t y of Eq. (2.1). We say that the a l g e b r a i c condition of s t r o n g p a r a b o t i c i t y is s a t i s f i e d if A c~l...C~m(~ ~ t ' ' ' r t , x), 13(~/~'''/3m, t, x) f o r any t, x, co a r e d i f f e r e n t i a b l e once with r e s p e c t to ~ e v e r y w h e r e except f o r a set having a finite n u m b e r of points of i n t e r s e c t i o n with e a c h line in the s p a c e of c o o r d i n a t e s ~, white the d e r i v a t i v e s a r e l o c a l l y s u m m a b l e on e a c h line in this s p a c e , and t h e r e e x i s t c o n s t a n t s a > 0, N such that f o r all ~ a t . . . a m , ~ l . . . ~ m , t, x, w,
1270
O/l [~z'''[~m [I:Y" 'Ym
+~
t, X) ~ . . . . . . m@3''' "iBm+[B
~
~''''~m(~v,...v% t, x) @ ~
1~ (2.9)
1~..... ~l~-~tn ...... m?..
I~zd+..-+[C~ml=m whe r e ~,...~ A...... ~
O As,. %~, 0~,...~ ~ ..
B ~''~m=
O B, 0~,-.-~m
and in the s e c o n d t e r m of t h e left s i d e of (2.9) s u m m a t i o n on ~1 . . . . , 6 m i s c a r r i e d out b e f o r e c o m p u t i n g the n o r m I "IQ. The n e x t t h e o r e m j u s t i f i e s the n a m e of c o n d i t i o n (2.9). T H E O R E M 2.2.
Suppose t h a t the a l g e b r a i c c o n d i t i o n of s t r o n g p a r a b o [ i e i t y [and a l l a s s u m p t i o n s of t h i s
s e c t i o n r e g a r d i n g A ( ~ f i l " " ~ m ) , B ( ( / 3 t ' " f l m ) ] is s a t i s f i e d . Proof,
Then Eq. (2.1) i s s t r o n g l y p a r a b o l i c .
We u s e the f o r m u l a of H a d a m a r c l ! 0
and a l s o the C a u c h y - B u n y a k o v s k i i i n e q u a l i t y and the d e f i n i t i o n of the o p e r a t o r s A , B.
We then obtain
I (~0I, :02) ~ 2 <<~01 - - :02' A (:01) - - A (~02)>> -}- I B (~01) - -
--B(~2)L~=- - 2 ( Dr - - A . . . . . . m( 0 8 ' ' "
. . , D+Zm (~1--~)2) ' A . . . . . . m i D ~' . . . D+m~ol)
D~m~~176 @ [11(B ( D f ~ ' . . . Dgm~ot) - - B (D ~ . . . D~v2)IQII~ ~< N t[ w~-- e2115 1
(2.10) f~,l+.-.+iC~ml~m
G
0
We have t h u s p r o v e d t h a t the c o n d i t i o n of m o n o t o n i e i t y (A 2) of S e e . II.2 is s a t i s f i e d . We have a c t u a l l y p r o v e d (A 2) with a r e s e r v e w h i c h e n a b i e s us to v e r i f y the c o e r c i v i t y c o n d i t i o n (A3): F r o m (2.10) and (2.7) we have (v~ = v , v~ = 0) I \ l[ f (t)lh 2<> + 1 B (v)]~ 4 1 (v, 0) + 2<>--I B (0)1~ + 21B (V)[Q[ B (0)[e ~< N ( ~[ ~-8)
1 + N 6 H v [[f,+2a [I v ][f,-b~-2 ii A(0)II~,.+N Hv H5 - ~ F2~-_~
.W,
[[ D~ ... D%nv H~,
]cq[+...§ :Zm[=m
w h e r e N d o e s not d e p e n d on 6, and 6 i s any n u m b e r g r e a t e r t h a n z e r o . U s i n g T h e o r e m s 1.4 a n d 1.5, n o t i n g t h a t the g r o w t h c o n d i t i o n (A 4) h a s a l r e a d y b e e n v e r i f i e d , and c h o o s i n g 5 s u f f i c i e n t l y s m a l l , we c o n c l u d e t h a t the c o e r c i v t t y c o n d i t i o n is s a t i s f i e d . The p r o o f of the t h e o r e m is c o m plete. Example.
L e t R d = E = R ~, and s u p p o s e t h a t Eq. (2.1) has the f o r m P
Om /I 0 m .tP-~ 0 'r~ (L x) = a (L ~o)( - - 1) ~+~ ~ kl ~ u (L ~t o-~
a'I c~"
~-dw
~(~' ~))d~+~(~, )~1o-~(~,~) 1
w h e r e a , b a r e a p p r o p r i a t e l y m e a s u r a b l e and a l s o b o u n d e d p r o c e s s e s . parabolicity become s
(0,
(2.1~)
H e r e the a l g e b r a i c c o n d i t i o n of s t r o n g
- - 2 ( p - - 1 ) a - ~ P@ b2~< - - ~ ,
w h e r e a > 0, ~ a c o n s t a n t . If t h i s c o n d i t i o n is s a t i s f i e d b y T h e o r e m 2.1 we o b t a i n a s s e r t i o n s r e g a r d i n g the e x i s t e n c e , u n i q u e n e s s , s t a b i l i t y with r e s p e c t to the i n i t i a l d a t a , and the M a r k o v p r o p e r t y of s o l u t i o n s of (2.1t) with the '~ooundary c o n d i t i o n " u E ~ (G). E q u a t i o n (2.11) e o i n c i d e s with (II.1.2). 3.
Cauchy
Problem
for
Linear
E q u a t i o n s of the f o r m (II.1.3) a r e s t u d i e d in the n e x t s e c t i o n .
Equations
of
Second
Order
tn t h i s s e c t i o n we continue the s t u d y of Eq. (2.1) a s s u m i n g t h a t m = 1, O = R d, p = 2, and A and B a r e l i n e a r f u n c t i o n s of ~ w h i c h a r e g e n e r a l l y n o t e q u a l to z e r o f o r ~ 0. M o r e o v e r , a l l a s s u m p t i o n s of S e e . 2 a r e n a t u r a l l y a s s u m e d to b e s a t i s f i e d .
1271
P r o b l e m (2.1)-(2.3) becomes
du(t, x)=D~(a~(t, x)D~tt(t, x)+f~(t, x))dt+(b=(t, x)D=tt(t, x)+g(t, x))dw(t)+dz(t, x), u(t, .)EL2(R~), it(0, x)=tto(X), xCR~, w h e r e aa~, f a a r e r e a l functions and b a and g a r e functions with values in E. alent to r e q u i r i n g the b o u n d e d n e s s of ac~, Ib~ IE and the inequality T
(3.1) (3.2)
Conditions (2.4), (2.5) are e q u i v -
T
M JIf. jfldt + M S lllgl ll d r < 0
0
A solution of p r o b l e m (3.1), (3.2) is u n d e r s t o o d in the sense of the i n t e g r a l identity (2.8); for a V - s o l u tion it is s a t i s f i e d f o r a l m o s t all (t, co), and f o r an H - s o l u t i o n it is s a t i s f i e d f o r e a c h t (a.s.). LEMMA 3.1.
Suppose that for a l l x , r/ e R d, t ~ [0, T], c9 (~2
i,j=I
i=1
w h e r e e is a c o n s t a n t , e > 0, aij = aa~, bi = b a , if a is the i-th and B the j - t h c o o r d i n a t e v e c t o r s . a l g e b r a i c condition of s t r o n g p a r a b o l i e i t y is satisfied. This [ e m m a is e a s i l y p r o v e d by m e a n s of inequalities of the type
Then the
2ao~l%~[ao~ I~ (~]~)2@s-t[a0~ ] (~0)2.
The next r e s u l t is a d i r e c t c o r o l l a r y of L e m m a 3.1, T h e o r e m s 2.1 and 2.2, and also C o r o l l a r y II.2.1. T H E O R E M 3.1. Suppose that condition (3.3) is s a t i s f i e d . Then t h e r e e x i s t s a function u(t, co) defined on [0, T] x ~ with values in L2(Rd), s t r o n g l y continuous in t in L2(Rd), :g-t - c o n s i s t e n t , and such that a) u ~ W~(R d) [ a . e . (t, co)l, r
b) m sup II. (t)ll + M t~T
il. r
dt <
~
0
c) f o r e a c h ~? ~ W~(Rd) t
(u (t), ~1)o= (uo, ~)o + j" ( - - 1)~ (D~u(s), 0
a~o (s) D~,])ods + f (-- 1)~ (f~
(s),
D~)ods
0
t
+ f (0~ (s) D ~ (s) + g (s), % dw (s) + (z (t), %
(3.4)
0
f o r all t ~ [0, T] (a. s.). T h e o r e m s 11.2.2 and 11.2.4 enable us to prove t h e o r e m s on the u n i q u e n e s s , stability with r e s p e c t to the initial data, and the M a r k o v p r o p e r t y of the function u c o n s i d e r e d in T h e o r e m 3.1 [u is an H - s o l u t i o n of p r o b lem (3.1), (3.2)]. Since they a r e p r o v e d s i m p l y by appealing to T h e o r e m s II.2.2, II.2.4, f o r b r e v i t y we shall not f o r m u l a t e these p r o p e r t i e s of u. We now turn to a m o r e i m p o r t a n t question - the question of r a i s i n g the s m o o t h n e s s of a solution of p r o b l e m (3.1), (3.2). The situation is that, e . g . , in the t h e o r y of filtration of diffusion p r o c e s s e s (see [25]) equations a n a l o g o u s to (3.4) a r i s e in which the s c a l a r p r o d u c t in L2(R d) is r e p l a c e d by the s c a l a r p r o d u c t in Wm(Rd), i . e . , the index 0 on the s c a l a r p r o d u c t in (3.4) is r e p l a c e d by m , w h e r e (% ~),~= (D~,... D%~q~, D~,... D ~ ) 0 . T h u s , we denote the m o d i f i e d equation (3.4) by (3.4)m. rn o r d e r that Eq. (3.4)m be meaningful f o r sufficiently s m o o t h u 0 and c o e f f i c i e n t s ao~, fo~, b u , g, z, it s u f f i c e s to r e s t r i c t attention to functions u belonging to wm+~(R d) [ a . e . (t, co)]. H o w e v e r , the filtration density is equal to the function (1 - A)mu m u l t i p l i e d by s o m e function of t i m e . T h u s , a s s e r t i o n s a r e n e e d e d r e g a r d i n g the m e m b e r s h i p of the solution of Eq. (3.4)m not in w m + I ( R d) but in W~an(Rd) [a.e. (t, co)]. H o w e v e r , m e r e l y an a s s e r t i o n on the e x i s t e n c e of a solution of (3.4) m with values in w~m(R d) is of little u s e , since f r o m the t h e o r y of filtration it is known a p r i o r i only that the solution b e l o n g s to wm+~(Rd); if we wish to p r o v e its s m o o t h n e s s , we m u s t have not only a t h e o r e m on the e x i s t e n c e of a solution of (3.4) m with values in w~m(R d) but also a t h e o r e m on the u n i q u e n e s s of a solution with values in wm+~(Rd). T h u s , a t h e o r e m on r a i s i n g the s m o o t h n e s s f o r Eq. (3.4)m is r e q u i r e d . A p r o o f of the
1272
c o r r e s p o n d i n g r e s u l t is given in [26]. In o r d e r not to o b s c u r e the exposition with t e c h n i c a l d e t a i l s , we h e r e p r o v e the t h e o r e m on r a i s i n g s m o o t h n e s s only f o r Eq. (3.4). T h r o u g h o u t the r e m a i n d e r of the p a p e r we a s s u m e that the i n t e g e r m >- 0, z(t) is a s q u a r e - i n t e g r a b [ e m a r t i n g a l e with values in W[ta(Rd) which is continuous in t in wrn(R d) f o r all t, co, the ftmctions ac[8(bce) have m d e r i v a t i v e s (weak d e r i v a t i v e s ) with r e s p e c t to x, a r e continuous (weakly continuous) in x, and these d e r i v a tives of a ~ , b(~ a r e bounded (for bc~ in the n o r m of E) u n i f o r m l y with r e s p e c t to t, x, w. Suppose that f o r all (t, w) the functions foz ~ Wm(Rd), u0 s Wm(Rd), and r
0
L e t g =- 0. This condition involves no t o s s of g e n e r a l i t y , since the i n t e g r a l of g with r e s p e c t to dw(t) can be included in z(t). F i n a l l y , we a s s u m e that condition (3.3) is satisfied. T H E O R E M 3.2. 3.1 b e l o n g s to w ~ n - ~ and
T h e r e e x i s t s a set 9.'ct~ such that P(f~') = 1 and f o r co 6 f~' the function u(t) of T h e o r e m and is continuous in t in the n o r m of Wm(Rd). M o r e o v e r , u e w m + I ( R d) [ a . e . (t, c~)], ?-
~I sup El./L~,~+ m j+ II" is)l%+1,#s < ~ . P r o o f . We set H = W ~ ( R d) and identify H with its dual by m e a n s of (" , ' ) m . It is then e a s i l y seen f r o m the P a r s e v a [ e~uality by m e a n s of this s a m e s c a l a r p r o d u c t that w~rt2-l(Rd) is identified with V*, the space dual to V = w~n+l(RU). T h u s , V ~ H ~ V * and it is obvious that e a c h imbedding is dense a~d eontinuous. As in the p r e c e d i n g s e e t i o n , it is e a s y to v e r i f y that the f o r m u l a s
((B (t) u) e, r,)~ =: ((b+,.it), e)eD+u. -,,)~ define bounded, l i n e a r o p e r a t o r s A(t) :V ~ V*, B(t)u : E ~ H. The r e a d e r can also v e r i f y without difficulty that the functions A (t)u, B (t)u s a t i s f y conditions (A t)-(A4) of Sec. II.2 and also the m e a s u r a b i l i t y conditions of this s e c t i o n (see, e . g . , [26]). H e n c e , by C o r o l l a r y I I . 2 . l in o u r e a s e Eq. (2.1) has an H - s o l u t i o n ~(t). shows that f o r e a c h rj e W m ( R d) t
(~(t). ~ ) ~ ( a o , j ~ q - S ( a ~ ( s )
Application of R e m a r k 2.1
f
D~[t(s)q-f(s),
(--1)~iD~'~),~ds+oI(b~(s)D~z(s),
0
~q),~dw(s)q-(z(~),r~),,
(3.5)
0
f o r all t ~ [0, T] (a. s.). To avoid m i s u n d e r s t a n d i n g s we note that (3.5) does not coincide with (3.4) m if m > 0 and as/?, b a depend on
X.
If in (3.5) in place of ~ we substitute (1 - A ) - m ~ and use the fact that by the P a r s e v a [ equality (f, g)m = (f, (1 - A)mg)0, w h e r e g ~ w~m(Rd), then we see that in (3.5) in place of m it is p o s s i b l e to write 0. A f t e r this, by T h e o r e m I[.2.2 we obtain sup I1u(~)-~ it)!1~_=0 (a. s.), and u(t) p o s s e s s e s the s a m e p r o p e r t i e s as ~(t). The p r o o f of the t h e o r e m is c o m p l e t e . E
h~ c o n c l u s i o n , we d i s c u s s the s i g n i f i c a n c e of condition (3.3) f o r the validity of T h e o r e m 3.1. R I we c o n s i d e r the equation Ou (t. x) dw(t), du (t, x)=-~- O'-~(t. ox ~x ) dt-:r-~ - -
w h e r e o- is a c o n s t a n t , with n o n r a n d o m initial data u0(x). (3.4) and the P a r s e v a t equality it follows that
If T h e o r e m 3.1 is valid f o r this equation, then f r o m
t
( ~ (t, ~)-;,:i~)d~ =- S ;;0(~)~7, i~) d ~ - - ,~- .
F o r d = 1,
t
.
.
.
.
.
.
.
.
t)
',i
w h e r e [ , ~ a r e the F o u r i e r t r a n s f o r m s of u, ~/. H e r e it is e a s y to i n t e r c h a n g e the i n t e g r a l s if ~(() is a cornpactly s u p p o r t e d function, and we then find that for a l m o s t all (t, ~, w) f
-
f
~ %~
(3.6) 0
0
12 73
We fix a [ for which Eq. (3.6) holds for a l m o s t all (t, co), and we denote the right side of (3.6) by ~o(t, 4). Then q)(t, ~) s a t i s f i e s (3.6) for all t (a.s.). The solution of the equation for ~p is known: 1
q0(t, D = e -T(~-~'n't-~i~w(~u0 ([). Since fi(t, 4) = (0(t, [) [a.e. (t, [ , w)], it follows that T ]1 - -
0
0 2
This implies that the left side of (3.7) is finite for all u 0 ~ L2(R 1) if and only if I cr[ < 1. The last condition in the p r e s e n t case is equivalent to (3.3). This example d e m o n s t r a t e s the n e c e s s i t y of condition (3.3) for the validity of T h e o r e m 3.1 and also the n e c e s s i t y of the c o e r e i v i t y condition (A 3) of Sec. II.2 for the validity of the r e s u l t s of Sec. II.2. LITERATURE I. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11.
12. 13. 14.
15. 16. 17. 18. 19. 20. 21.
1274
CITED
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87. 88. 89.
TIME OF
ASYMPTOTICS
EVOLUTION
NUMBER R.
OF L.
OF
FOR
SOME
SYSTEMS
DEGENERATE
WITH
AN
MODELS
INFINITE
PARTICLES Dobrushin
and
Yu.
M.
Sukhov*
UDC 519.219
The paper is devoted to the problem of convergence to the equilibrium state in the motion of infinite s y s t e m s of c l a s s i c a l p a r t i c l e s . Two models of the motion are considered: free motion of point particles in Euclidean spaee R u, u -> 1, and motion of solid rods on the line R ~. The paper contains new r e s u l t s obtained by the authors and also a survey of previous r e s u l t s in this d i r e c tion. I.
Introduction
The subject of classical equilibrium statistical mechanics is the study of probabilistic characteristics of a large system of interacting particles in the equilibrium state, i.e., after a sufficiently long, autonomous evolution of the system. The main postulate of statistical mechanics formulated by Boltzmann and Gibbs asserts that such characteristics are described by means of probability distributions of special type which have received the name of Gibbs equilibrium distributions. The Boltzmann - Gibbs postulate is still very far from justification at a mathematical level. The traditional approach to this, which is connected with the well-lmown ergodic hypothesis for a system of a finite number of mechanical particles (see, e.g., [2, 48] and the bibliography presented there), does not seem so promising today. This is illustrated, in particular, by the fact that the modern theory of dynamical systems (the theory of Kolmogorov-Arnol'd-Moser; see [1, 35, 50, 84, 85]) has shown that the ergodic hypothesis is not true at [east in part of the physically natural situations. Another approach to the mathematical justification of the Boltzmann - Gibbs postulate has become popular in recent years. This approach is based on considering a system consisting of an infinite number of interacting particles rather than a large finite system. This makes it possible to describe clearly and simply laws only approximately apparent in a large finite system. The "infinite-particle" approach has proved its fruitfulness in application to equilibrium statistical mechanics where the Gibbs states of an infinite system of particles are studied, i.e., Gibbs probability measures on phase space (in probability-theoretic terminology - Gibbs random fields). A survey of this topic can be found, e.g., in [39, 41, 77, 83], the works [33, 70], and in the literature cited in these publications. In c o r r e s p o n d e n c e with generally adopted physical ideas it is natural to generally understand by equiiibrium states of an infinite s y s t e m of particles the Gibbs states defined by a potential describing the motion of the particles and depending on three additional p a r a m e t e r s : two s c a l a r p a r a m e t e r s of t e m p e r a t u r e and e h e m i cal potential and the v e c t o r p a r a m e t e r of the mean m o m e n t u m of the p a r t i c l e s . The p r e s e n c e of additional p a r a m e t e r s is connected with the existence of three " c l a s s i c a l " integrals of the motion of a finite s y s t e m : the total energy, the n u m b e r of p a r t i c l e s , and the total momentum. These p a r a m e t e r s given the particle density, the specific total e n e r g y , and the specific velocity of p a r t i c l e s in the equilibrium state. A natural conjecture (see, e . g . , [I0, 11, 12, 53, 59])is that in the general case of physically r e a l , nondegenerate potentials the three * K. Boldrigini took part in the work on the paper. T r a n s l a t e d f r o m ltogi Nauki i Tekhniki, Seriya Sovremennye P r o b l e m y Matematiki, Vol. 14, pp, 147-254, 1979.
0090-4104/81/1604-1277 $0%50 9 1981 Plenum Publishing Corporation
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