Stochastic evolution equations in UMD Banach spaces

Abstract

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)) dW_H(t), t\in [0,\Tend], 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 $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathcal{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is $\F_0$-measurable and bounded, then this problem has a unique mild solution, which has trajectories in $C^ł([0,T];\D((-A)^θ)$ provided $\lambda \ge 0$ and $\theta \ge 0$ satisfy $ł+\theta <\frac{1}{2}$. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations.

Accepted for publication in Journal of Functional Analysis