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: $$W_2(μ_t,μ_\infty)^2 +{\rm Ent}(μ_t|μ_\infty)\le c {\rm e}^{-λt} \min\big\{W_2(μ_0, μ_\infty)^2,{\rm Ent}(μ_0|μ_\infty)\big\},\ \ t\ge 1,$$ where $c,λ>0$ are constants, $μ_t$ is the distribution of the solution at time $t$, $μ_\infty$ 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.

Theorems & Definitions (15)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• proof : Proof of Theorem \ref{['T0']}
• proof : Proof of Theorem \ref{['C2.0']}
• Lemma 4.1
• proof