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. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process ${\displaystyle {\left\{W_t^H\right\}}_{t\in [0,T]}}$ with Cameron-Martin 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) ${\displaystyle d{X}_t=A{X}_{t}dt+Bd{W}_t^H}$ (t∈ [0,T]), ${\displaystyle X_0=0}$ almost surely, where A is the generator of a ${\displaystyle C_0}$-semigroup ${\displaystyle {\left\{S\left(t\right)\right\}}_{t\ge 0}}$ 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 ${\displaystyle X_t={\int }^t\_\left\{0\right\}S(t-s)Bd{W}_s^H}$.