Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem
Let H be a separable real Hilbert space and let E be a separable real Banach space. In this paper we develop a general theory of stochastic convolution of L(H,E)− valued functions with respect to a cylindrical Wiener process {WH t }t∈[0,T ] with CameronMartin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP ) dXt = AXt dt+B dW H t (t ∈ [0, T ]), X0 = 0 almost surely, where A is the generator of a C0−semigroup {S(t)}t≥0 of bounded linear operators on E and B ∈ L(H,E) is a bounded linear operator. We further show that whenever a weak solution exists, it is unique, and given by a stochastic convolution