On the stability of the invariant probability measures of McKean-Vlasov equations

TL;DR

This paper investigates the long-term stability of invariant measures in McKean-Vlasov equations, providing new criteria using Lions derivatives and analyzing both local and non-local cases on Euclidean space and the torus.

Contribution

It introduces novel stability criteria for McKean-Vlasov equations utilizing Lions derivatives and analyzes stability on different geometries with explicit convergence rates.

Findings

Stability determined by roots of an analytic function on .

Fourier coefficients of the interaction kernel influence stability.

Exponential convergence in Wasserstein metric W_1 achieved.

Abstract

We study the long-time behavior of some McKean-Vlasov stochastic differential equations used to model the evolution of large populations of interacting agents. We give conditions ensuring the local stability of an invariant probability measure. Lions derivatives are used in a novel way to obtain our stability criteria. We obtain results for non-local McKean-Vlasov equations on $\mathbb{R}^d$ and for McKean-Vlasov equations on the torus where the interaction kernel is given by a convolution. On $\mathbb{R}^d$, we prove that the location of the roots of an analytic function determines the stability. On the torus, our stability criterion involves the Fourier coefficients of the interaction kernel. In both cases, we prove the convergence in the Wasserstein metric $W_1$ with an exponential rate of convergence.