1.1. Basic concepts This part of our article is devoted to the contraction of Frechet smooth sto- chastic seminows for mild and weak solutions of semilinear see's and spde's. In Theorem 1.2.6, it is shown that mild solutions of semilinear see's in a Hilbert space H generate smooth perfect locally compacting cocycles. The construction of the cocycle for semilinear see's is based on the following new strategy, which bypasses the need for Kolmogorov's continuity theorem: We "lift" the linear terms of the see to the Hilbert-Schmidt operators L2(H). We represent the mild solution of the linear see as a "chaos-type" series ex- pansion living in the Hilbert space L2(H) of Hilbert-Schmidt operators on H (Theorems 1.2.1-1.2.3). Using a variational technique, the mild solution of the full semilinear see is represented in terms of the linear cocycle constructed above (Theorems 1.2.1- 1.2.4). This part of the strategy requires the non-linear part of the see to satisfy a Lipschitz condition (Theorem 1.2.6). Similar variational techniques are used to construct smooth cocycles for weak solutions of specific classes of spde's. In particular, we obtain smooth stochastic semiflows for semilinear spde's driven by cylindrical Brownian motion. In these applications, it turns out that in addition to smoothness of the non-linear terms, one requires some level of dissipativity or Lipschitz continuity, e.g. the stochastic heat equation (Theorem 1.3.5), the stochastic reaction diffusion equation (Theo- rems 1.4.1, 1.4.2) and stochastic Burgers equation with additive infinite-dimensional noise (Theorem 1.4.3). We begin by formulating the ideas of a stochastic semiflow and a cocycle which are central to the analysis in this work. Let (Q, T, P) be a probability space. Denote by f the P-completion of T, and let (O,^*, (^i)t0j-P) De a complete filtered probability space satisfying the usual conditions ([Pr]). Denote A := {(s,t) G R 2 : 0 s t}, and R + := [0,oo). For a topological space E, let 8(E) denote its Borel cr-algebra. Let H e a positive integer and 0 e 1. If E and N are real Banach spaces, we will denote by L^k\E, N) the Banach space of all /c-multilinear maps A : Ek —• N with the uniform norm ||^4|| := s\xp{\A(vi,V2r " v k)\ : Vi G E, \vi\ l,i = 1, , k}. Suppose U C E is an open set. A map / : U N is said to be of class Cfe'e if it is Ck and if D^ f : U -* L(k\E,N) is e-H61der continuous on bounded sets in U. A Ck,e map / : U N is said to be of class Cb'e if all its derivatives D^f,l j k, are globally bounded on [/, and D^ f is e-H61der continuous on U. A mapping / : [0,T] x U * iV is of class Ck'e in the second variable uniformly with respect to the first if for each t G [0, T], /(£, •) is Ck,e on [/, for every bounded set UQ C U the spatial partial derivatives D^f(t,x),j = 1, , k, are uniformly bounded in (£, x) G [0, T] x J7o and the corresponding e-H61der constant of D^f(t, -)\U0 is uniformly bounded in t G [0,T]. The following definitions are crucial to the developments in this article. DEFINITION 1.1.1. Let E be a Banach space, k a non-negative integer and e G (0,1]. A stochastic Ck,e semiflow on E is a random field V:AxExQ-^E satisfying the following properties: (i) V is {8(A) 0 8(E) 0 T, B(E))-measurable. (ii) For each wGO, the map A x E 3 (s, t, x) i— V(s, t, x,u) G E is continuous. 6
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