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 C0C0-semigroup on a UMD Banach space E   and WHWH 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 u0u0 is F0F0-measurable and bounded, then this problem has a unique mild solution, which has trajectories in Cλ([0,T];D(θ(−A))))Cλ([0,T];D((−A)θ))) provided λ⩾0λ⩾0 and θ⩾0θ⩾0 satisfy λ+θ<12. Various extensions are given and the results are applied to parabolic stochastic partial differential equations.