Stochastic evolution equations in UMD Banach spaces

By J. V. Neerven, Mark Veraar, Lutz Weis. 2008

185 citations recorded by the source corpus.

We discuss existence, uniqueness, and space-time Holder 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 2 (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 C ((0,T);D(( A) ) provided 0 and 0 satisfy + < 1 2 . Vari- ous extensions of this result are given and the results are applied to parabolic stochastic partial dierential equations.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/0bb7a12e7409adcfb1875b7e691e01fb9632fdfa

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Citations in the corpus

γ-Radonifying Operators: A Survey (2010, unknown): “, Brzeźniak [11, 13], Da Prato and Zabczyk [27], van Neerven, Veraar, Weis [86, 89] and the references therein.”
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Stochastic delay fractional evolution equations driven by fractional Brownian motion (2014, background): “[3], Taniguchi [4], Jentzen and Röckner [5], Ren et al.”
Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient (2017, background): “We refer for instance to [10, 49, 50] for further properties.”
General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions (2012, background): “[21, 22]) does not fit the present case because, if a Banach space is UMD, then it is reflexive.”
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