Calculus of Variations and Geometric Measure Theory

L. Ambrosio - G. Savaré - L. Zambotti

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

created by ambrosio on 19 Apr 2007
modified by savare on 12 Jan 2009


Accepted Paper

Inserted: 19 apr 2007
Last Updated: 12 jan 2009

Journal: Probability Theory and Related Fields
Year: 2007


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.