Asymptotic Couplings by Reflection and Applications for Non-Linear Monotone SPDES

Asymptotic couplings by reflection are constructed for a class of non-linear monotone SPDES (stochastic partial differential equations). As applications, the gradient/H\"older estimates as well as the exponential convergence are derived for the associated Markov semigroup. The main results are illustrated by stochastic generalized porous media equations, stochastic $p$-Laplacian equations, and stochastic generalized fast-diffusion equations. We emphasize that the gradient estimate is studied at the first time for these equations.