  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.
