Ergodicity for stochastic T-monotone parabolic obstacle problems
Yassine Tahraoui
TL;DR
This work analyzes ergodicity for stochastic obstacle problems governed by a nonlinear T-monotone operator in a bounded domain with homogeneous boundaries and multiplicative noise. It constructs a Markov-Feller semigroup on $K_\psi$ via a penalization approach, and proves the existence (and under dissipativity, uniqueness and mixing) of ergodic invariant measures using Krylov–Bogoliubov and Krein–Milman, with Lewy-Stampacchia inequalities to control the reflection measure. The analysis covers both $2\le p<\infty$ and $\underline{p}<p<2$ regimes, providing moment bounds and convergence rates to equilibrium, and extends invariant-measure results beyond the 1D/additive-noise setting to higher-dimensional, nonlinear, stochastic obstacle problems. The results are significant for theory and applications where obstacle constraints, parabolic nonlinear operators (including the $p$-Laplacian), and multiplicative noise interact, enabling rigorous long-time behavior descriptions and uniqueness under natural dissipativity conditions.
Abstract
This work aims to investigate the existence of ergodic invariant measures and its uniqueness, associated with obstacle problems governed by a T-monotone operator defined on Sobolev spaces and driven by a multiplicative noise in a bounded domain of $\mathbb{R}^d$ with homogeneous boundary conditions. We show that the solution defines a Markov-Feller semigroup defined on the space of real bounded continuous functions of a convex subset related to the obstacle and we prove the existence of ergodic invariant measures and its uniqueness, under suitable assumptions. Our method relies on a combination of "Krylov Bogoliubov theorem", "Krein Milman theorem" and Lewy-Stampacchia inequalities to control the reflection measure.