Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space–time Hölder regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t)+F(t,U(t))) dt +B(t,U(t)) dWH (t ), t ∈ [0,T0], U(0) = u0, 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 F0-measurable and bounded, then this problem has a unique mild solution, which has trajectories in Cλ([0,T ];D((−A)θ ))) provided λ 0 and θ 0 satisfy λ +θ < 12 . Various extensions are given and the results are applied to parabolic stochastic partial differential equations.