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

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