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: W 2 ( μ t , μ ∞ ) 2 + Ent ( μ t | μ ∞ ) ≤ c e − λ t min { W 2 ( μ 0 , μ ∞ ) 2 , Ent ( μ 0 | μ ∞ ) } , t ≥ 1 , where c , λ > 0 are constants, μ t is the distribution of the solution at time t , μ ∞ is the unique invariant probability measure, Ent is the relative entropy and W 2 is the L 2 -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.