Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations

François Bolley, Ivan Gentil, Arnaud Guillin

Published 2011-10-17, updated 2012-09-18Version 2

We describe conditions on non-gradient drift diffusion Fokker-Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, which to our knowledge is a novelty.