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<p>This work aims to investigate the existence of ergodic invariant measures and its uniqueness, associated with obstacle problems governed by a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">T</mi> </math></EquationSource> </InlineEquation>-monotone operator defined on Sobolev spaces and driven by a multiplicative noise in a bounded domain of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> 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.</p>

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Ergodicity for stochastic T-monotone parabolic obstacle problems

  • 期刊论文

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  • 发表日期：

    2026年3月26日

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  • Yassine Tahraoui

摘要

This work aims to investigate the existence of ergodic invariant measures and its uniqueness, associated with obstacle problems governed by a \(\textbf{T}\) T -monotone operator defined on Sobolev spaces and driven by a multiplicative noise in a bounded domain of \(\mathbb {R}^d\) 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.