We discuss existence, uniqueness, and space-time H\"older regularity for
solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) +
F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where A
generates an analytic C0-semigroup on a UMD Banach space E and WH is a
cylindrical Brownian motion with values in a Hilbert space H. We prove that
if the mappings F:[0,T]×E→E and B:[0,T]×E→L(H,E) satisfy suitable Lipschitz conditions and u0 is
\F_0-measurable and bounded, then this problem has a unique mild solution,
which has trajectories in C^\l([0,T];\D((-A)^\theta) provided λ≥0
and θ≥0 satisfy \l+\theta<\frac12. Various extensions of this
result are given and the results are applied to parabolic stochastic partial
differential equations.Comment: Accepted for publication in Journal of Functional Analysi