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

TL;DR

This paper establishes the existence and uniqueness of solutions for stochastic porous media equations on general measure spaces with non-continuous diffusivity functions, broadening the scope of previous models to include complex spaces and operators.

Contribution

It generalizes stochastic porous media equations to measure spaces with non-continuous monotone diffusivity functions and non-local operators, without requiring coercivity.

Findings

Proved existence and uniqueness of solutions in broad settings.

Developed an $L^p(\mu)$-It\^o formula of independent interest.

Applicable to fractals, manifolds, and non-local operators like Bernstein functions.

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.