Ergodicity for stochastic T-monotone parabolic obstacle problems

TL;DR

This paper establishes the existence and uniqueness of ergodic invariant measures for stochastic obstacle problems governed by T-monotone operators, using advanced probabilistic and functional analysis techniques.

Contribution

It introduces a novel approach combining Krylov Bogoliubov, Krein Milman, and Lewy-Stampacchia inequalities to analyze ergodicity in obstacle problems with multiplicative noise.

Findings

Existence of ergodic invariant measures proven.

Uniqueness of these measures established.

Solution defines a Markov-Feller semigroup.

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.