Stochastic evolution equations with nonlinear diffusivity, recent progress and critical cases
Ioana Ciotir, Dan Goreac, Jonas M. Tölle
TL;DR
The paper surveys stochastic evolution equations with nonlinear diffusivity, focusing on critical parameter regimes and boundary cases. It casts the SPDEs in variational form with drift $A(u)=\Delta \Psi(u)$ and noise $B(u)\,dW(t)$, and discusses multiple solution concepts including variational, mild/strong, martingale, and stochastic variational inequality (SVI) formulations. It collects results on existence, uniqueness, convergence under graph-convergence of $\Psi$, homogenization, and ergodicity, illustrating with porous medium, fast diffusion, singular $p$-Laplace, and total variation flow models. The work highlights connections to self-organized criticality, nonlinear diffusion with gradient or Stratonovich noise, and provides a roadmap for regularity and numerical analysis in these critical settings.
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.