Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations

Benjamin Gess, Michael Röckner

Type: Article

Publication Date: 2016-06-01

Citations: 21

DOI: https://doi.org/10.1090/tran/6981

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.

Locations

arXiv (Cornell University) - View - PDF
Transactions of the American Mathematical Society - View

Similar Works

Action	Title	Year	Authors
+	Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations	2014	Benjamin Gess Michael Röckner
+	Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations	2014	Benjamin Gess Michael Röckner
+ PDF Chat	Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE	2018	Benjamin Gess Martina Hofmanová
+	REGULARIZATION BY STOCHASTIC DRIFT	2012	Paul-Éric Chaudru de Raynal
+	Regularity of Stochastic Kinetic Equations	2016	E. Fedrizzi Franco Flandoli Enrico Priola Julien Vovelle
+	Regularity of Stochastic Kinetic Equations	2016	E. Fedrizzi Franco Flandoli Enrico Priola Julien Vovelle
+	Regularity of stochastic kinetic equations	2017	E. Fedrizzi Franco Flandoli Enrico Priola Julien Vovelle
+	Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness	2019	Lisa Beck Franco Flandoli Massimiliano Gubinelli Mario Maurelli
+	Stochastic continuity equations with conservative noise	2017	Benjamin Gess Scott M. Smith
+	Stochastic continuity equations with conservative noise	2017	Benjamin Gess Scott A. Smith
+ PDF Chat	Strong existence and uniqueness for stochastic differential equation with Hölder drift and degenerate noise	2017	Paul-Éric Chaudru de Raynal
+	Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE	2016	Benjamin Gess Martina Hofmanová
+	Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE	2016	Benjamin Gess Martina Hofmanová
+ PDF Chat	On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values	2021	Arnulf Jentzen Benno Kuckuck Thomas Müller-Gronbach Larisa Yaroslavtseva
+ PDF Chat	Weak regularization by stochastic drift : Result and counter example	2018	Paul-Éric Chaudru de Raynal
+	Nonlinear parabolic stochastic evolution equations in critical spaces Part II. Blow-up criteria and instantaneous regularization	2020	Antonio Agresti Mark Veraar
+	Nonlinear parabolic stochastic evolution equations in critical spaces Part II. Blow-up criteria and instantaneous regularization	2020	Antonio Agresti Mark Veraar
+	Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence	2020	Antonio Agresti Mark Veraar
+ PDF Chat	Nonlinear parabolic stochastic evolution equations in critical spaces part II	2022	Antonio Agresti Mark Veraar
+	Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise	2023	Yushi Hamaguchi

Works That Cite This (19)

Action	Title	Year	Authors
+	Stability and moment estimates for the stochastic singular Φ-Laplace equation	2023	Florian Seib Wilhelm Stannat Jonas M. Tölle
+	Stability and moment estimates for the stochastic singular $Φ$-Laplace equation	2021	Florian Seib Wilhelm Stannat Jonas M. Tölle
+	Nonlinear diffusion equations with nonlinear gradient noise	2020	Konstantinos Dareiotis Benjamin Gess
+ PDF Chat	Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients	2021	Mengyu Cheng Zhenxin Liu
+	Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions	2019	Jonas M. Tölle
+	Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation	2021	Florian Seib Wilhelm Stannat Jonas M. Tölle
+ PDF Chat	Singular-Degenerate Multivalued Stochastic Fast Diffusion Equations	2015	Benjamin Gess Michael Röckner
+ PDF Chat	Variational solutions to nonlinear stochastic differential equations in Hilbert spaces	2018	Viorel Barbu Michael Röckner
+ PDF Chat	Doubly nonlinear stochastic evolution equations	2020	Luca Scarpa Ulisse Stefanelli
+ PDF Chat	Refined existence and regularity results for a class of semilinear dissipative SPDEs	2020	Carlo Marinelli Luca Scarpa

Works Cited by This (18)

Action	Title	Year	Authors
+	Introduction to the Theory of (Non-Symmetric) Dirichlet Forms	1992	Zhi-Ming Ma Michael Röckner
+ PDF Chat	Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow	2012	Abdelhadi Es–Sarhir Max‐K. von Renesse Wilhelm Stannat
+ PDF Chat	Uniqueness of motion by mean curvature perturbed by stochastic noise	2004	Nung Kwan Yip Panagiotis E. Souganidis
+	Fully nonlinear stochastic pde with semilinear stochastic dependence	2000	Pierre‐Louis Lions Panagiotis E. Souganidis
+	Fully nonlinear stochastic partial differential equations	1998	Pierre‐Louis Lions Panagiotis E. Souganidis
+ PDF Chat	Finite element approximations of the stochastic mean curvature flow of planar curves of graphs	2014	Xiaobing Feng Yukun Li Andreas Prohl
+ PDF Chat	Pairings between measures and bounded functions and compensated compactness	1983	Gabriele Anzellotti
+	A stochastic selection principle in case of fattening for curvature flow	2001	Nicolas Dirr Stephan Luckhaus Matteo Novaga
+ PDF Chat	The Euler equation for functionals with linear growth	1985	Gabriele Anzellotti
+ PDF Chat	A Parabolic Quasilinear Problem for Linear Growth Functionals	2002	F. Andreu Vicent Caselles José M. Mazón