Stability of solutions to stochastic partial differential equations

TL;DR

This paper develops a comprehensive framework to analyze the stability of solutions to stochastic partial differential equations (SPDEs) under perturbations, using stochastic variational inequalities and convergence concepts.

Contribution

It introduces the stochastic variational inequalities framework for stability analysis of SPDEs and applies it to various models including stochastic fast diffusion and p-Laplace equations.

Findings

Established continuous dependence of solutions on convex potential perturbations.

Derived Trotter type and homogenization results for specific stochastic PDEs.

Proved convergence of nonlocal to local stochastic p-Laplace models.

Abstract

We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. In particular, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. The generality of the developed framework is then laid out by deducing Trotter type and homogenization results for stochastic fast diffusion and stochastic singular p-Laplace equations. In addition, we provide an SVI treatment for stochastic nonlocal p-Laplace equations and prove their convergence to the respective local models.