Exponential Convergence of Non-Linear Monotone SPDEs

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 $C\in (0,\mathrm{\infty })$ such that $$\underset{\mu (f^2)\le 1}{\sup }\mathrm{\parallel }{P}_{t}f-\mu (f){\mathrm{\parallel }}_{\mathrm{\infty }}\le C{\text{e}}^{-\ll t},t\ge 1.$$ 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.