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

Benjamin Gess; Jonas Tölle

doi:10.1137/15M1049774

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

Benjamin Gess, Jonas Tölle

Department of Mathematics and Systems Analysis

Research output: Contribution to journal › Article › Scientific › peer-review

5 Citations (Scopus)

99 Downloads (Pure)

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 language	English
Pages (from-to)	4094-4125
Journal	SIAM Journal on Mathematical Analysis
Volume	48
Issue number	6
DOIs	https://doi.org/10.1137/15M1049774
Publication status	Published - 8 Dec 2016
MoE publication type	A1 Journal article-refereed

Keywords

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

Access to Document

10.1137/15M1049774

15m1049774Final published version, 345 KB

https://arxiv.org/abs/1507.04545

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Gess, B., & Tölle, J. (2016). Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations. SIAM Journal on Mathematical Analysis, 48(6), 4094-4125. https://doi.org/10.1137/15M1049774

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title = "Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations",

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. T{\"o}lle, J. Math. Pures Appl., 101 (2014), pp. 789--827], [W. Liu and J. M. T{\"o}lle, 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. R{\"o}ckner, 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.",

keywords = "Stochastic variational inequality, Nonlocal stochastic partial differential equations, Singular-degenerate SPDE, stochastic $p$-Laplace equation, ergodicity",

author = "Benjamin Gess and Jonas T{\"o}lle",

year = "2016",

month = "12",

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doi = "10.1137/15M1049774",

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journal = "SIAM Journal on Mathematical Analysis",

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T1 - Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations

AU - Gess, Benjamin

AU - Tölle, Jonas

PY - 2016/12/8

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N2 - 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.

AB - 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.

KW - Stochastic variational inequality

KW - Nonlocal stochastic partial differential equations

KW - Singular-degenerate SPDE

KW - stochastic $p$-Laplace equation

KW - ergodicity

UR - https://arxiv.org/abs/1507.04545

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DO - 10.1137/15M1049774

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EP - 4125

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JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

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