Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions

Let $k\in (d,\mathrm{\infty }]$ and consider the $k\ast$-distance $$\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{k\ast }:=\sup \{\mathrm{\mid }\mu (f)-\nu (f)\mathrm{\mid }:f\in {\text{B}}_b({\mathbb{R}}^d),\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathtt{L}}^{\mathtt{k}}}:=\underset{x\in {\mathbb{R}}^d}{\sup }\mathrm{\parallel }{1}_{B(x,1)}f{\mathrm{\parallel }}_{L^k}\le 1\}$$ between probability measures on ${\mathbb{R}}^d$. The exponential ergodicity in $1$-Wasserstein and $k\ast$ distances is derived for a class of McKean-Vlasov SDEs with small singular interactions measured by $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{k\ast }\mathrm{.}$ Moreover, the exponential ergodicity in $2$-Wasserstein distance and relative entropy is derived when the interaction term is given by $$b^{(0)}(x,\mu ):={\int }_{{\mathbb{R}}^d}h(x-y)\mu (\text{d}y)$$ for some measurable function $h:{\mathbb{R}}^d\to {\mathbb{R}}^d$ with small $\mathrm{\parallel }h{\mathrm{\parallel }}_{{\mathtt{L}}^{\mathtt{k}}}$.