This is an announcement for the paper "Stochastic evolution equations in UMD Banach spaces" by J.M.A.M. van Neerven, M.C. Veraar, and L. Weis.

Abstract: We discuss existence, uniqueness, and space-time H"older regularity for solutions of the parabolic stochastic evolution equation [\left{\begin{aligned} dU(t) & = (AU(t) + F(t,U(t))),dt + B(t,U(t)),dW_H(t), \qquad t\in [0,\Tend],\ U(0) & = u_0, \end{aligned} \right. ] 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 $B:[0,T]\times E\to \mathscr{L}(H,E)$ 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 $C^\l([0,T];\D((-A)^\theta)$ provided $\lambda\ge 0$ and $\theta\ge 0$ satisfy $\l+\theta<\frac12$. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations.

Archive classification: math.FA math.PR

Mathematics Subject Classification: 47D06; 60H15; 28C20; 46B09

Remarks: Accepted for publication in Journal of Functional Analysis

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Submitted from: mark@profsonline.nl

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