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

TL;DR

This paper proves ergodicity for both local and nonlocal stochastic p-Laplace equations across all dimensions and p-values in [1,2), extending previous work and solving an open problem for the total variation flow.

Contribution

It establishes ergodicity without spatial restrictions and demonstrates convergence of invariant measures from nonlocal to local stochastic p-Laplace equations.

Findings

Ergodicity proven for all p in [1,2) and any spatial dimension.

Includes the multivalued case of stochastic total variation flow.

Shows convergence of invariant measures under rescaling.

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 [Gess, T\"{o}lle; J. Math. Pures Appl., 2014], [Liu, T\"{o}lle; Electron. Commun. Probab., 2011], [Liu; J. Evol. Equations, 2009]. In particular, the results include the multivalued case of the stochastic (nonlocal) total variation flow, which solves an open problem raised in [Barbu, Da Prato, R\"{o}ckner; SIAM J. Math. Anal., 2009]. 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.