Convergence of invariant measures for singular stochastic diffusion equations

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 $\mathrm{\Lambda }\subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters $p$ and $r$ respectively (in the Hilbert spaces $L^2(\mathrm{\Lambda })$, $H^{-1}(\mathrm{\Lambda })$ respectively). The highly singular limit case $p=1$ is treated with the help of stochastic evolution variational inequalities, where $\text{mathbbm}P$-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).