In this paper we study the following non-autonomous stochastic evolution
equation on a UMD Banach space E with type 2,
{equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t)))
dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],
U(0) & = u_0. {aligned}. {equation}
Here (A(t))t∈[0,T] are unbounded operators with domains
(D(A(t)))t∈[0,T] which may be time dependent. We assume that
(A(t))t∈[0,T] satisfies the conditions of Acquistapace and Terreni. The
functions F and B are nonlinear functions defined on certain interpolation
spaces and u0∈E is the initial value. WH is a cylindrical Brownian
motion on a separable Hilbert space H.
Under Lipschitz and linear growth conditions we show that there exists a
unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation
spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk,
to obtain space-time regularity results for the solution U of
\eqref{eq:SEab}. For Hilbert spaces E we obtain a maximal regularity result.
The results improve several previous results from the literature.
The theory is applied to a second order stochastic partial differential
equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to
several improvements of their result.Comment: Accepted for publication in Journal of Evolution Equation