Wasserstein-Łojasiewicz inequalities and asymptotics of McKean-Vlasov equation
Beomjun Choi, Seunghoon Jeong, Geuntaek Seo
TL;DR
This work addresses convergence to equilibrium for the nonconvex McKean–Vlasov equation on the flat torus by developing a Wasserstein Łojasiewicz–Simon framework that requires only analyticity of the confinement and interaction potentials. The authors pull back the energy to the tangent space at a stationary measure and prove an analytic gradient map and a Fredholm second variation, enabling a finite-dimensional Łojasiewicz reduction and a gradient-flow convergence result without log-Sobolev or convexity assumptions. They derive explicit convergence rates depending on a Łojasiewicz exponent and show the Keller–Segel system on the torus, among other radial-interaction models, fit within this framework. The results significantly extend convergence theory for Wasserstein gradient flows to genuinely nonconvex settings and suggest broad applicability to other nonconvex gradient-flow problems.
Abstract
We prove convergence to equilibrium for solutions to the McKean-Vlasov (granular media) equation on the flat torus in a genuinely nonconvex setting. Our approach is based on a Wasserstein-Łojasiewicz gradient inequality for the associated free energy, established under mild analyticity assumptions on the confinement and interaction potentials; this yields convergence of the corresponding Wasserstein gradient flow without any convexity assumptions and without postulating log-Sobolev related functional inequalities. We expect this strategy to extend to more general nonconvex Wasserstein gradient flows; in the present work we develop it in the McKean-Vlasov setting, with the Keller-Segel chemotaxis model on the torus as a key application.