Asymptotic Couplings by Reflection and Applications for Non-Linear Monotone SPDES

TL;DR

This paper develops asymptotic couplings by reflection for non-linear monotone SPDEs, leading to gradient and exponential convergence estimates, with novel first-time gradient analysis for these equations.

Contribution

It introduces a new coupling method for non-linear monotone SPDEs and derives gradient and convergence estimates, including the first gradient study for these equations.

Findings

Gradient/Hölder estimates established for the first time.

Exponential convergence of the associated Markov semigroup demonstrated.

Applications to various stochastic PDEs like porous media and p-Laplacian equations.

Abstract

Asymptotic couplings by reflection are constructed for a class of non-linear monotone SPDES (stochastic partial differential equations). As applications, the gradient/H\"older estimates as well as the exponential convergence are derived for the associated Markov semigroup. The main results are illustrated by stochastic generalized porous media equations, stochastic $p$-Laplacian equations, and stochastic generalized fast-diffusion equations. We emphasize that the gradient estimate is studied at the first time for these equations.