Existence and Stability for Fokker-Planck equations with log-concave reference measure

TL;DR

This paper establishes existence, stability, and convergence results for Markov processes linked to Fokker-Planck equations with log-concave measures, using entropy and Wasserstein geometry, with applications to SPDEs and interface models.

Contribution

It proves a general existence theorem and a stability property for Markov processes associated with nonlinear Fokker-Planck equations involving convex functionals.

Findings

Existence of Markov processes under convexity assumptions.

Weak convergence of invariant measures implies process convergence.

Applications to stochastic PDEs and interface fluctuation models.

Abstract

We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker-Planck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces.