Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p -Laplace equation, p ∈ ( 1 , 2 ) and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r ∈ ( 0 , 1 ) on a bounded open domain Λ ⊂ 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 ( Λ ) , H − 1 ( Λ ) respectively). The highly singular limit case p = 1 is treated with the help of stochastic evolution variational inequalities, where P -a.s. convergence, uniformly in time, is established. shown that the associated unique invariant measures of the ergodic semigroups converge in the weak sense (of probability measures).