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

The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(\mu_t,\mu_\infty)^2 +{\rm Ent}(\mu_t|\mu_\infty)\le c {\rm e}^{-\lambda t} \min\big\{W_2(\mu_0, \mu_\infty)^2,{\rm Ent}(\mu_0|\mu_\infty)\big\},\ \ t\ge 1,$$ 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, ${\rm Ent}$ 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.