Exponential Convergence of Non-Linear Monotone SPDEs

Abstract

For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\ll >0$ 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 $$ \sup_{μ(f^2)\le 1}\|P_tf-μ(f)\|_\infty \le C \e^{-\ll t},\ \ t\ge 1.$$ 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.

19 pages

References in corpus (3)

• Harnack inequality and applications for stochastic generalized porous media equations
• Local and global well-posedness of SPDE with generalized coercivity conditions
• Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts

Cited by in corpus (2)

• Strong Feller Property and Irreducibility for Non-Linear Monotone SPDEs
• Asymptotic Couplings by Reflection and Applications for Non-Linear Monotone SPDES