The following type of exponential convergence is proved for (non-degenerate or degenerate) McKean–Vlasov SDEs: ${\mathbb{W}}_2{({\mathrm{\mu }}_t,{\mathrm{\mu }}_{\infty })}^2+\mathrm{Ent}({\mathrm{\mu }}_t\left|{\mathrm{\mu }}_{\infty }\right)\le c{\mathrm{e}}^{-\lambda t}min\{{\mathbb{W}}_2{({\mathrm{\mu }}_0,{\mathrm{\mu }}_{\infty })}^2,\mathrm{Ent}\left({\mathrm{\mu }}_0\right|{\mathrm{\mu }}_{\infty })\},t\ge 1,$ where $c,\lambda >0$ are constants, ${\mathrm{\mu }}_t$ is the distribution of the solution at time $t$, ${\mathrm{\mu }}_{\infty }$ is the unique invariant probability measure, $\mathrm{Ent}$ is the relative entropy and ${\mathbb{W}}_2$ 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.

对于（非退化的或退化的）McKean–Vlasov SDE，证明了以下类型的指数收敛： ${\mathbb{w\; ^}}_2{（{\mathrm{\mu }}_{Ť}，{\mathrm{\mu }}_{\infty }）}^2+\mathrm{恩特}（{\mathrm{\mu }}_{Ť}|{\mathrm{\mu }}_{\infty }）\le C{\mathrm{Ë}}^{-\lambda Ť}分\{{\mathbb{w\; ^}}_2{（{\mathrm{\mu }}_0，{\mathrm{\mu }}_{\infty }）}^2，\mathrm{恩特}（{\mathrm{\mu }}_0|{\mathrm{\mu }}_{\infty }）\}，Ť\ge \mathrm{1个}，$ 哪里 $C，\lambda >0$ 是常数 ${\mathrm{\mu }}_{Ť}$ 是当时解决方案的分布 $Ť$， ${\mathrm{\mu }}_{\infty }$ 是唯一不变的概率测度， $\mathrm{恩特}$ 是相对熵， ${\mathbb{w\; ^}}_2$ 是个 ${\mathrm{大号}}^2$-Wasserstein距离。特别是，这种类型的指数收敛适用于某些（非退化的或退化的）粒状介质类型方程，这些方程概括了Carrillo等人研究的方程。（2003）和Guillin等。（0000）关于平均场熵的指数收敛。