Exponential convergence of non-linear monotone SPDEs

Feng-Yu Wang

doi:10.3934/dcds.2015.35.5239

Abstract

For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\lambda>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that $$ \sup_{\mu(f^2)\le 1}||P_tf-\mu(f)||_\infty \le C e^{-\lambda t}, t\ge 1.$$ By using the coupling by change of measure in the line of [F.-Y. Wang, Ann. Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential convergence rate are derived for a class of non-linear monotone stochastic partial differential equations. The main result is illustrated by the stochastic porous medium equation and the stochastic $p$-Laplace equation respectively. Finally, the $V$-uniformly exponential convergence is investigated for stochastic fast-diffusion equations.

Highlights

It is well known that the solution to the porous medium equation (1.1)dXt = ∆Xtr dt decays at the algebraic rate t− 1 r−1 as t → ∞, where ∆ is the DirichletLaplacian on a bounded domain in Rd, r > 1 is a constant and Xtr := |Xt|r−1Xt, see [1]
L2(m), see Section 3), the equation has a unique solution starting at x, and the associated Markov semigroup Pt has a unique invariant probability measure μ such that
We note that in [5] an lower bound estimate of exponential convergence rate is presented for a class of semi-linear SPDEs

Summary

Introduction

Laplacian on a bounded domain in Rd, r > 1 is a constant and Xtr := |Xt|r−1Xt, see [1]. This type algebraic convergence has been extended in [3] to stochastic generalized porous media equations. We note that in [5] an lower bound estimate of exponential convergence rate is presented for a class of semi-linear SPDEs (stochastic partial differential equations). We aim to present explicit lower bound estimates for the ultra-exponential convergence rate λ in (1.3). We prove a general result for a class of non-linear SPDEs considered in [7].

A general result

Stochastic porous medium equation

Stochastic p-Laplace equation

Exponential convergence for stochastic fast-diffusion equations