Ergodicity for stochastic T-monotone parabolic obstacle problems

2023/11/05 by Tahraoui, Yassine
#35K86 #35R35 #47D07 #60H15 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2311.02637

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 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.

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