Stochastic Porous Media Equation on General Measure Spaces with Increasing Lipschitz Nonlinearties

Abstract

We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E,\mathcal{B}(E),\mu )$, and the Laplacian replaced by a self-adjoint operator $L$. In the case of Lipschitz nonlinearities $\Psi$, we in particular generalize previous results for open $E\subset {\mathbb{R}}^d$ and $L=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed initial data and in avoiding the restriction to superlinear behavior of $\Psi$ at infinity for $L^2(\mu )$-initial data.

23 pages, revised version, (H2)(iii) added, some misprints corrected, Claim 3.1 changed with more details added