Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions

TL;DR

This paper establishes exponential ergodicity for McKean-Vlasov SDEs with singular interactions, using novel distance measures and conditions on the interaction kernel, advancing understanding of long-term behavior in complex stochastic systems.

Contribution

It introduces new ergodicity results for McKean-Vlasov SDEs with singular interactions measured by a $k*$-distance, extending prior work to more general interaction kernels.

Findings

Exponential ergodicity in 1-Wasserstein and $k*$ distances for small singular interactions.

Exponential ergodicity in 2-Wasserstein distance and relative entropy under specific interaction forms.

Conditions on the interaction kernel's $L^k$ norm ensure ergodic behavior.

Abstract

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