Ergodicity and Local Limits for Stochastic Local and Nonlocal p-Laplace Equations

Ergodicity for local and nonlocal stochastic singular p-Laplace equations is proven, without restriction on the spatial dimension and for all p ∈ [1, 2). This generalizes previous results from [Gess, Tölle; JMPA, 2014], [Liu, Tölle; ECP, 2011], [Liu; JEE, 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, 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.