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

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Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations. / Gess, Benjamin; Tölle, Jonas.

In: SIAM Journal on Mathematical Analysis, Vol. 48, No. 6, 08.12.2016, p. 4094-4125.

Research output: Contribution to journalArticleScientificpeer-review

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Gess, Benjamin ; Tölle, Jonas. / Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations. In: SIAM Journal on Mathematical Analysis. 2016 ; Vol. 48, No. 6. pp. 4094-4125.

@article{b45eed306bf942bf8d9f57094e9448c7,
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",
day = "8",
doi = "10.1137/15M1049774",
language = "English",
volume = "48",
pages = "4094--4125",
journal = "SIAM Journal on Mathematical Analysis",
issn = "0036-1410",
number = "6",

}

TY - JOUR

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

AU - Gess, Benjamin

AU - Tölle, Jonas

PY - 2016/12/8

Y1 - 2016/12/8

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

U2 - 10.1137/15M1049774

DO - 10.1137/15M1049774

M3 - Article

VL - 48

SP - 4094

EP - 4125

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 6

ER -

ID: 9576465