Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations

Gess, Benjamin; Röckner, Michael

doi:10.1090/tran/6981

Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
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Trans. Amer. Math. Soc. 369 (2017), 3017-3045 Request permission

Abstract:

The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous noise. In particular, this provides a characterization of solutions to such SPDE in terms of (generalized) strong solutions. Second, for the one-dimensional stochastic mean curvature flow with normal noise we adapt the notion of stochastic variational inequalities to provide a characterization of solutions previously obtained in a limiting sense only. This solves a problem left open by A. Es-Sarhir and M.-K. von Renesse in 2012 and sharpens regularity properties obtained by them with W. Stannat.

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