Abstract of a paper by J.M.A.M. van Neerven, M.C. Veraar, and L. Weis
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 The source file(s), scp_arxiv.tex: 157532 bytes, is(are) stored in gzipped form as 0804.0932.gz with size 44kb. The corresponding postcript file has gzipped size 241kb. Submitted from: mark@profsonline.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.0932 or http://arXiv.org/abs/0804.0932 or by email in unzipped form by transmitting an empty message with subject line uget 0804.0932 or in gzipped form by using subject line get 0804.0932 to: math@arXiv.org.
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