Exponential Convergence in Entropy and Wasserstein Distance for McKean-Vlasov SDEs

Abstract

The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2({\mu }_t,{\mu }_{\infty }{)}^2+\mathrm{Ent}({\mu }_t|{\mu }_{\infty })\le c{\mathrm{e}}^{-\lambda t}\min \{W_2({\mu }_0,{\mu }_{\infty }{)}^2,\mathrm{Ent}({\mu }_0|{\mu }_{\infty })\},t\ge 1,$$ where $c,\lambda >0$ are constants, ${\mu }_t$ is the distribution of the solution at time $t$, ${\mu }_{\infty }$ is the unique invariant probability measure, $\mathrm{Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy.

25 pages