Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation

TL;DR

This paper investigates the stability, long-term behavior, and moment bounds of stochastic singular $\

Contribution

It introduces improved moment estimates and convergence rates for stochastic $\\Phi$-Laplace equations, extending previous results and providing new concentration and dissipativity insights.

Findings

Enhanced moment estimates for stochastic $\\Phi$-Laplace equations

Quantitative convergence rates to invariant measures

New concentration results and maximal dissipativity of the Kolmogorov operator

Abstract

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular stochastic $p$-Laplace equations and, more generally, singular stochastic $\Phi$-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir, von Renesse and Stannat, NoDEA 16(9), 2012, and improve the results by Liu and T\"olle, ECP 16, 2011.