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 Banac h space. We develop a general theory of stochastic convolution of L(H, E)-v alued functions with respect to a cylindrical Wiener process {W-t(H)}(t is an element of [0,T]) with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak sol ution of the stochastic abstract Cauchy problem dX(t) = AX(t)dt + BdW(t)(H) (t is an element of [0, T]), (ACP) X-0 = 0 almost surely, where A is the generator of a C-0-semigroup {S(t)}(t greater than or equal to0) of bounded linear operators on E and B is an element of L(N, E) is a b ounded linear operator. We further show that whenever a weak solution exist s, it is unique, and given by a stochastic convolution x(t) = (0)integral (t) S(t - s)B dW(s)(H).