Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: a variational approach

TL;DR

This paper establishes the existence of invariant measures and solvability of Kolmogorov equations for a class of dissipative stochastic PDEs with singular drift, using a variational approach to handle non-standard growth conditions.

Contribution

It introduces a variational framework to prove invariant measures and Kolmogorov equation solvability for SPDEs with singular, non-growth-restricted drift terms.

Findings

Existence of invariant measures for the semigroup generated by the SPDE.

Solvability of the Kolmogorov equation in $L^1(u)$ space.

Identification of the generator as the closure of the Kolmogorov operator.

Abstract

We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures $\mu$, solvability of the associated Kolmogorov equation in $L^1(\mu)$ is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting.