Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time Holder 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 is an element of [0,T-0], U(0) = u(0), where A generates an analytic C-0-semigroup on a UMD, Banach space E and W-H is a cylindrical Brownian motion with values in a Hilbert space H. We prove that if the mappings F: [0, T] x E -> E and B: [0, T] x E -> L(H, E) satisfy suitable Lipschitz conditions and uo is F-0-measurable and bounded, then this problem has a unique mild solution, which has trajectories in C-lambda ([0, T]; D((-A)(theta)))) provided lambda >= 0 and theta >= 0 satisfy lambda + theta < 1/2. Various extensions are given and the results are applied to parabolic stochastic partial differential equations. (c) 2008 Elsevier Inc. All rights reserved.