Exponential convergence in entropy and Wasserstein for McKean–Vlasov SDEs

The following type of exponential convergence is proved for (non-degenerate or degenerate) McKean–Vlasov SDEs: W2(?t,??)2+Ent(?t|??)?ce??tmin{W2(?0,??)2,Ent(?0|??)},t?1, where c,?>0 are constants, ?t the distribution solution at time t, ?? unique invariant probability measure, Ent relative entropy and W2 L2-Wasserstein distance. In particular, this holds some granular media equations generalizing those studied in Carrillo et al. (2003) Guillin (0000) on a mean field entropy.