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 [Gess, Tölle; J. Math. Pures Appl., 2014], [Liu, Tölle; 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, Röckner; 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.

30 pages; to appear in SIAM J. Math. Anal

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Cited by in corpus (6)

• Stability of solutions to stochastic partial differential equations
• An averaged space-time discretization of the stochastic $p$-Laplace system
• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
• Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organised criticality
• Ergodicity for singular-degenerate porous media equations
• Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation