Stochastic evolution equations in UMD Banach spaces

Abstract. We discuss existence, uniqueness, and space-time Hölder regular-ity 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 < 1 2. Vari-ous extensions of this result are given and the results are applied to parabolic stochastic partial differential equations