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Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations


Authors: Benjamin Gess and Michael Röckner
Journal: Trans. Amer. Math. Soc. 369 (2017), 3017-3045
MSC (2010): Primary 60H15; Secondary 35R60, 35K93
DOI: https://doi.org/10.1090/tran/6981
Published electronically: July 15, 2016
MathSciNet review: 3605963
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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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Additional Information

Benjamin Gess
Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
Email: bgess@mis.mpg.de

Michael Röckner
Affiliation: Faculty of Mathematics, University of Bielefeld, 33615 Bielefeld, Germany
Email: roeckner@mathematik.uni-bielefeld.de

DOI: https://doi.org/10.1090/tran/6981
Keywords: Degenerate SPDE, degenerate $p$-Laplace, mean curvature flow, regularity, stochastic variational inequalities, linear growth functionals
Received by editor(s): May 23, 2014
Published electronically: July 15, 2016
Article copyright: © Copyright 2016 American Mathematical Society