Strong Feller property, irreducibility, and uniqueness of the invariant measure for stochastic PDEs with degenerate multiplicative noise

TL;DR

This paper proves strong Feller and irreducibility properties for certain stochastic PDEs with degenerate multiplicative noise, ensuring uniqueness of invariant measures without strong dissipativity assumptions.

Contribution

It introduces a novel method to analyze the effects of degeneracy in noise on the regularity and ergodicity of solutions to nonlinear SPDEs.

Findings

Established strong Feller property for degenerate noise cases.

Proved irreducibility and uniqueness of invariant measures.

Provided a new approach to handle degeneracy near potential barriers.

Abstract

We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under no strong-dissipativity assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. We propose a method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the degenerate noise coefficient. From the mathematical perspective, this is one of the first contributions in the literature establishing strong Feller properties and irreducibility in the multiplicative degenerate case, and opens up novel investigation paths in the direction of regularisation effects and ergodicity in the degenerate-noise framework. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type.