Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

We study regularity properties for invariant measures of semilinear diffusions in a separable Hilbert space. Based on a pathwise estimate for the underlying stochastic convolution, we prove a priori estimates on such invariant measures. As an application, we combine such estimates with a new technique to prove the $L^1$-uniqueness of the induced Kolmogorov operator, defined on a space of cylindrical functions. Finally, examples of stochastic Burgers equations and thin-film growth models are given to illustrate our abstract result.Comment: 19 page

Existence and uniqueness of invariant measures for a class of transition semigroups on Hilbert spaces

AbstractWe prove a smoothing property and the irreducibility of transition semigroups corresponding to a class of semilinear stochastic equations on a separable Hilbert space H. Existence and uniqueness of invariant measures are discussed as well

Maximal dissipativity of Kolmogorov operators with Cahn–Hilliard type drift term

AbstractWe prove the existence of invariant measures μ for Kolmogorov operators LF associated with semilinear stochastic partial differential equations with Cahn–Hilliard type drift term. Based on gradient estimates on the pseudo-resolvent associated with LF and a priori estimates for the moments of μ we prove maximal dissipativity of LF in the space L1(μ)

Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow

We establish moment estimates for the invariant measure of a stochastic partial differential equation describing motion by mean curvature flow in (1+1) dimension, leading to polynomial stability of the associated Markov semigroup. We also prove maximal dissipativity for the related Kolmogorov operator