Convergence of invariant measures for singular stochastic diffusion equations

Abstract

It is proved that the solutions to the singular stochastic $p$-Laplace equation, $p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter $r\in (0,1)$ on a bounded open domain $\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters $p$ and $r$ respectively (in the Hilbert spaces $L^2(\Lambda )$, $H^{-1}(\Lambda )$ respectively). The highly singular limit case $p=1$ is treated with the help of stochastic evolution variational inequalities, where $\mathbbm{P}$-a.s. convergence, uniformly in time, is established. It is shown that the associated unique invariant measures of the ergodic semigroups converge in the weak sense (of probability measures).

to appear in Stoch. Proc. Appl. (in press), 18 pp

References in corpus (6)

• On ergodicity of some Markov processes
• Stochastic Porous Media Equation and Self-Organized Criticality
• Multi-valued, singular stochastic evolution inclusions
• Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts
• Ergodicity of Stochastic Curve Shortening Flow in the Plane
• Convergence of solutions to the $p$-Laplace evolution equation as $p$ goes to 1

Cited by in corpus (7)

• Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise
• Multi-valued, singular stochastic evolution inclusions
• Stability of solutions to stochastic partial differential equations
• Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations
• Corrigendum to `Convergence of invariant measures for singular stochastic diffusion equations'
• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
• Improved regularity for the stochastic fast diffusion equation