Ergodicity for stochastic T-monotone parabolic obstacle problems

Abstract This work aims to investigate the existence of ergodic invariant measures and its uniqueness, associated with obstacle problems governed by a $$\textbf{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> -monotone operator defined on Sobolev spaces and driven by a multiplicative noise in a bounded domain of $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> 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.