Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises (Q2189791)

The authors study asymptotic properties of the Markovian strong Feller semigroups associated to stochastic Cahn-Hilliard equations in one space dimension with singular logarithmic nonlinearity and additive Gaussian noise. The solutions to the stochastic equation take values in \([-1,1]\) due to the implementation of two reflection measures, comparable to the situation in [\textit{A. Debussche} and the first author, SIAM J. Math. Anal. 43, No. 3, 1473--1494 (2011; Zbl 1233.60036)]. The results extend those from [the first author and \textit{L. Manca}, Stochastic Processes Appl. 125, No. 10, 3785--3800 (2015; Zbl 1321.60134)], where the asymptotic strong Feller property of the semigroup associated to a similar equation was studied. The main results include the existence of an exponentially integrable invariant measure for highly degenerate colored noise that satisfy the essentially elliptic condition, and an asymptotic log-Harnack inequality, which is derived via coupling methods. Following the ideas of [\textit{F.-Y. Wang}, Ann. Probab. 39, No. 4, 1449--1467 (2011; Zbl 1238.60069)] for the case of non-degenerate space-time white noise, the authors prove a dimension-free Harnack inequality for the Markov semigroup, which then yields heat kernel estimates for the transition densities.