Stochastic evolution equations with nonlinear diffusivity, recent progress and critical cases

TL;DR

This survey reviews recent advances in stochastic evolution equations with nonlinear diffusion, focusing on critical cases, solution concepts, convergence, homogenization, and recent progress in regularity, long-term behavior, and numerical methods.

Contribution

It summarizes recent progress on critical stochastic evolution equations, highlighting solution notions, convergence results, and developments in regularity and ergodicity.

Findings

Analysis of solution convergence depending on parameters

Progress in regularity and long-time behavior studies

Advances in numerical analysis of stochastic equations

Abstract

This short survey article stems from recent progress on critical cases of stochastic evolution equations in variational formulation with additive, multiplicative or gradient noises. Typical examples appear as the limit cases of the stochastic porous medium equation, stochastic fast- and super fast-diffusion equations, self-organized criticality, stochastic singular $p$-Laplace equations, and the stochastic total variation flow, among others. We present several different notions of solutions, results on convergence of solutions depending on a parameter, and homogenization. Furthermore, we provide some references hinting at the recent progress in regularity results, long-time behavior, ergodicity, and numerical analysis.