Exponential convergence in entropy and Wasserstein for McKean–Vlasov SDEs

Abstract

The following type of exponential convergence is proved for (non-degenerate or degenerate) McKean–Vlasov SDEs: W2(μt,μ∞)2+Ent(μt|μ∞)≤ce−λtmin{W2(μ0,μ∞)2,Ent(μ0|μ∞)},t≥1, where c,λ>0 are constants, μt is the distribution of the solution at time t, μ∞ is the unique invariant probability measure, Ent is the relative entropy and W2 is the L2-Wasserstein distance. In particular, this type of exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in Carrillo et al. (2003) and Guillin et al. (0000) on the exponential convergence in a mean field entropy.