THE STOCHASTIC SEMIFLOW U(tU L0) U(t2, 0(tuw)) U(h +t2,x,,uj) fi 8(h,-) 0(h,-) t = h +12 FIGURE 1. The Cocycle Property. (a) Linear see's We will first prove the existence of semiflows associated with mild solutions of linear stochastic evolution equations of the form: (1.2.1) du{t, x, •) = — Au(t, x, -)dt + Bu(t, x, •) dW{t), U(0,X,UJ) =x G H. t0] In the above equation A : D(A) C H —* H is a closed linear operator on a sep- arable real Hilbert space H. Assume that A has a complete orthonormal system of eigenvectors {en : n 1} with corresponding positive eigenvalues {iin,n 1} i.e., Aen = /i n e n , n 1. Suppose —A generates a strongly continuous semigroup of bounded linear operators Tt : H — i7, t 0. Let E be a separable Hilbert space and W(t),t 0, be an E-valued Brownian motion with a separable covari- ance Hilbert space K, and defined on the canonical complete filtered Wiener space (fi, T, (^rt)to5 P), satisfying the usual conditions. Here K C E is a Hilbert-Schmidt

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