1.2. FLOWS AND COCYCLES OF SEMILINEAR SEE'S 7 (hi) For fixed (s, t, uS) G A x fi, the map ^ 3 i ^ X(s, £, x,cu) e E is Cfc,e. (iv) If 0 r s t, a G fJ and x G £, then V(r, £, x, a ) = V(s, t, F(r, s, x, a ), a ). (v) For all (s, x, a ) G R + x E x ( ] , one has V (5, 5, x, a ) = x. DEFINITION 1.1.2. Let 0 : R x Q £1 be a P-preserving (S(R) 0 ^ , J7)- measurable group on the probability space (SI,,?1*, P), E1 a Banach space, k a non- negative integer and e G (0,1]. A Ck,e perfect cocycle (U,9) on E is a (£(R+) ® JB(25) (8) T, S(E))-measurable random field U : R + x E xfl ^ E with the following properties: (i) For each W G O , the map R + x E 3 (£, x) i-» £/(£, x, a ) G E is continuous and for fixed (£,u ) G R + x fi, the map E 3 x i- U(t,x,uj) e E is Cfe,e. (ii) {/(£ + 5,., a ) - U(t, •, 0(s, a )) o U(s, •, a ) for all s, £ G R + and all u G Q. (hi) [7(0, x, a ) = x for all x G £ , u G fi. Note that a cocycle (£/, 0) corresponds to a one-parameter semigroup on E x(], viz. R + x £ x f t - £ x n (t,(x,u ))^([/(t,x,u ),^(t,u )) Fig.l illustrates the cocycle property. The vertical solid lines represent random copies of E sampled according to the probability measure P. The main objective of this part of our article is to show that under sufficient regularity conditions on the coefficients, a large class of semilinear see's and spde's admits a Ch,e semiflow F : A x i J x f 2 ^ i 7 f o r a suitably chosen state space H with the following property: For every x G H, y(£o,-,x, •) coincides a.s. for all t to with the mild/weak solution of the see/spde with initial function x at t = to. In the autonomous case, we show further that the semiflow V generates a cocycle (U,0) on if, in the sense of Definition 1.1.2 above. The cocycle and its Frechet derivative are compact in all cases. 1.2. Flows and cocycles of semilinear see's In this section, we will establish the existence and regularity of semiflows gen- erated by mild solutions of semilinear see's. We will begin with the linear case. In fact, the linear cocycle will be used to represent the mild solution of the semilinear see via a variational formula which transforms the semilinear see to a random inte- gral equation (Theorem 1.2.5). The latter equation plays a key role in establishing the regularity of the stochastic flow of the semilinear see (Theorem 1.2.6). One should note at this point the fact that Kolmogorov's continuity theorem fails for random fields parametrized by infinite-dimensional spaces. As a simple example, consider the random field / : L2([0,1],R) L2(f2,R) defined by the Wiener integral I(x):= [ x(t)dW(t), X G L 2 ( [ 0 , 1 ] , R ) , Jo where W is one-dimensional Brownian motion. The above random field has no continuous (or even linear!) measurable selection L2([0,1],R) x Q R ([Mo.l], pp. 144-148 [Mo.2]).
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