Stochastic Generalized Porous Media Equations over $\sigma$-finite Measure Spaces with Non-Continuous Diffusivity Function

Abstract

In this paper, we prove that stochastic porous media equations over $\sigma$-finite measure spaces $(E,\mathcal{B},\mu)$, driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator $L$ and the diffusivity function given by a maximal monotone multi-valued function $\Psi$ of polynomial growth, have a unique solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions $\Psi$, for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an $L^p(\mu)$-It\^{o} formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular SOC models) and cases where $E$ is a manifold or a fractal, and to non-local operators $L$, as e.g. $L=-f(-\Delta)$, where $f$ is Bernstein function.