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

TL;DR

This paper establishes existence and uniqueness of solutions for stochastic porous media equations on general measure spaces, extending previous results to fractional Laplacians and removing several restrictive assumptions.

Contribution

It generalizes prior work by allowing fractional Laplacians, removing transience assumptions, and broadening initial data and nonlinearity conditions.

Findings

Proved existence and uniqueness of solutions.

Extended results to fractional Laplacians.

Dropped transience assumption on operator L.

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 $(E, \mathscr{B}(E), \mu)$, 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.