Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations

François Bolley (1) , Ivan Gentil (2) , Arnaud Guillin (3, 4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 ICJ - Institut Camille Jordan
3 LMBP - Laboratoire de Mathématiques Blaise Pascal
4 IUF - Institut universitaire de France

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.

Mots clés: Diffusion equations; Wasserstein distance; functional inequalities; spectral gap

Domaines: Probabilités [math.PR]

François Bolley, Ivan Gentil, Arnaud Guillin. Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations. Journal of Functional Analysis, 2012, 263 (8), pp.2430-2457. ⟨10.1016/j.bbr.2011.03.031⟩. ⟨hal-00632941v2⟩