Convergence of invariant measures for singular stochastic diffusion equations

Details Convergence of invariant measures for singular stochastic diffusion equations Convergence of invariant measures for singular stochastic diffusion equations

2012-01-13

arxiv.org/abs/1201.2839v1

Mathematics

Probability

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

Scientific paper

10.1016/j.spa.2011.11.011

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).

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