Convergence of invariant measures for singular stochastic diffusion equations

TL;DR

This paper proves the continuous dependence of solutions to certain singular stochastic PDEs on parameters and shows the weak convergence of their invariant measures, including the highly singular case $p=1$, using stochastic evolution variational inequalities.

Contribution

It establishes parameter-dependent continuity of solutions and invariant measures for singular stochastic PDEs, including the challenging case $p=1$, with new methods involving stochastic evolution variational inequalities.

Findings

Solutions are continuous in mean with respect to parameters p and r.

Invariant measures converge weakly as parameters vary.

Strong convergence results for the case p=1 using variational inequalities.

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 $\Lambda\subset\R^d$ 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).