Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations

Benjamin Gess, Jonas Tölle

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Abstract

Ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in[1,2)$. This generalizes previous results from [B. Gess and J. M. Tölle, J. Math. Pures Appl., 101 (2014), pp. 789--827], [W. Liu and J. M. Tölle, Electron. Commun. Probab., 16 (2011), pp. 447--457], [W. Liu, J. Evol. Equations, 9 (2009), pp. 747--770]. In particular, the results include the multivalued case of the stochastic (nonlocal) total variation flow, which solves an open problem raised in [V. Barbu, G. Da Prato, and M. Röckner, SIAM J. Math. Anal., 41 (2009), pp. 1106--1120]. Moreover, under appropriate rescaling, the convergence of the unique invariant measure for the nonlocal stochastic $p$-Laplace equation to the unique invariant measure of the local stochastic $p$-Laplace equation is proven.
Original languageEnglish
Pages (from-to)4094-4125
JournalSIAM Journal on Mathematical Analysis
Volume48
Issue number6
DOIs
Publication statusPublished - 8 Dec 2016
MoE publication typeA1 Journal article-refereed

Keywords

  • Stochastic variational inequality
  • Nonlocal stochastic partial differential equations
  • Singular-degenerate SPDE
  • stochastic $p$-Laplace equation
  • ergodicity

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