Exponential convergence of non-linear monotone SPDEs

Feng-Yu  Wang

doi:10.3934/dcds.2015.35.5239

Article Contents

2015, Volume 35, Issue 11: 5239-5253. Doi: 10.3934/dcds.2015.35.5239

This issue Previous Article Large deviation principle for stochastic heat equation with memory Next Article Stochastic Korteweg-de Vries equation driven by fractional Brownian motion

Exponential convergence of non-linear monotone SPDEs

Feng-Yu Wang^1,

1.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875

Received: October 2013

Revised: October 2014

Published: November 2015

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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.

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Mathematics Subject Classification: Primary: 60H155; Secondary: 60B10.

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