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...
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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~ 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.).
1239
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 - 0, then M s u p l x s]2- 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]-.- 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
~=~ 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
~; 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![Alis+ 0
i [~(s)12d < m ) ~< c~,
0
w he r e co
B o t h t h e s e c o n d i t i o n s a r e s a t i s f i e d , s i n c e 1](s) 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 ... 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, 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)]Pd(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,,sup 1k v (t ?)I 9~,~ , t J t Z ,7
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 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 ...
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 - 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 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~b2~0, 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 - 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~~ 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-. 0 ( a . s . ) T
F
I s.p
It, x), A It/
_ 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 ~ 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 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~ 0 such that f o r any u E Wp (G)
IlUllm.,-.> -}- 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