Ergodicity for Stochastic Porous Media Equations with Multiplicative Noise

Abstract

The long time behavior of solutions to stochastic porous media equations with nonlinear multiplicative noise on bounded domains with Dirichlet boundary data is studied. Based on weighted L 1-estimates, the existence and uniqueness of invariant measures with optimal bounds on the rate of mixing are proved. Along the way, the existence and uniqueness of entropy solutions are shown.

Ergodicity for Singular-Degenerate Stochastic Porous Media Equations

“…[39]), treating weakly dissipative drift operators. Stochastic porous media equations with purely multiplicative noise have been treated in [21], using a dissipativity estimate in a weighted L 1 space. Lower bound techniques as introduced in [35,36] were used by [31] and [32], where generalized porous media equations with discontinuous nonlinearities are analyzed as explained before.…”

Finite time extinction for the 1D stochastic porous medium equation with transport noise

“…To this end, the proof loosely follows the strategy of Dareiotis, Gerencsér and Gess [7] or Dareiotis and Gess [9] who study entropy solutions for stochastic porous medium type equations posed on the torus. The Cauchy-Dirichlet problem in the framework of entropy solutions was recently studied in the work of Dareiotis, Gess and Tsatsoulis [10]. However, they do not consider the case of gradient type noise.…”

“…Since we have to incorporate Brownian paths as the lateral boundary, the following constructions are of course random. However, all results of this section turn out to be purely deterministic consequences of the probabilistic facts (8), (9), (10) and (11). In other words, we proceed with constructions to be understood in a pathwise sense.…”

“…So fix an integer M ≥ 1 as well as some κ > 0. Let C α be the square integrable random variable of the estimate (10). Let finally ε > 0 be fixed, and denote by ε = ε (M, ε) the constant from (11).…”

“…In the work [7], the authors consider the case of nonlinear multiplicative noise with periodic boundary conditions. The subsequent work of Dareiotis and Gess [9] treats the regime of nonlinear conservative gradient noise (again with periodic boundary conditions), whereas the recent work [10] by Dareiotis, Gess and Tsatsoulis provides the corresponding theory of [7] for the Dirichlet problem on smooth domains.…”