Homogenization and Long Time Behavior of Wasserstein gradient flow for McKean-Vlasov Equation

Abstract

We consider a McKean-Vlasov equation incorporating spatial fast oscillatory behavior into both the energetic and kinematic components of the model. The former is given in terms of a confining potential while the latter by an underlying metric. The equation is formulated as a gradient flow in the space of probability measures endowed with a Wasserstein metric. We identify the limiting dynamics which is also described as a gradient flow in an effective media. We further prove the long time exponential convergence of solutions to the invariant measure. The rate of convergence is expressed in terms of a Logarithmic Sobolev constant which is uniform in terms of the oscillatory length scale. Our approach is based on variational and functional inequalities. The homogenization result of the current paper extends the authors'previous work [GaoYip] on a linear Fokker-Planck equation to a nonlinear and nonlocal setting.