Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions

Abstract

Let $k\in (d,\infty ]$ and consider the $k\ast$-distance $$\|μ-ν\|_{k*}:= \sup\Big\{|μ(f)-ν(f)|:\ f\in\B_b(\R^d),\ \|f\|_{\tt L^k}:=\sup_{x\in \R^d}\|1_{B(x,1)}f\|_{L^k}\le 1\Big\}$$ 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 $\Vert \cdot {\Vert }_{k\ast }.$ Moreover, the exponential ergodicity in $2$-Wasserstein distance and relative entropy is derived when the interaction term is given by $$b^{(0)}(x,μ) :=\int_{\R^d}h(x-y)μ(\d y)$$ for some measurable function $h:{\mathbb{R}}^d\to {\mathbb{R}}^d$ with small $\Vert h{\Vert }_{{\mathtt{L}}^{\mathtt{k}}}$.

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