Convergence of invariant measures for singular stochastic diffusion equations
For singular stochastic \(p\)-Laplace equations with \(p\in(1,2)\) and for stochastic fast diffusion equations with parameter~\(r\in(0,1)\) on a bounded open domain in \(\mathbb R^d\), with a cylindrical Wiener process, continuous dependence of solutions in mean on \(p\) and \(r\), resp., uniformly in time from \((0,T)\), in the corresponding Hilbert spaces is shown. Also continuous dependence of the unique invariant measures of the induced Markov semigroups on \(p\) and on \(r\), resp., in the topology of weak convergence is derived. For \(p=1\) and \(d=1\) or \(d=2\), using another notion of a solution, \(P\)-almost sure strong convergence of solutions for \(p\), \(p>1\), is shown for \(p\to 1\), uniformly in \([0,T]\).
scientific article; zbMATH DE number 6040311
scientific article