An optimal transport approach of hypocoercivity for the 1d kinetic Fokker-Plank equation

TL;DR

This paper introduces a new quadratic optimal transport metric tailored for the 1D kinetic Fokker-Planck equation, providing novel estimates for convergence to equilibrium even with non-convex potentials.

Contribution

It develops a new quadratic transport metric that is equivalent to Wasserstein-2 and establishes trend to equilibrium estimates for the kinetic Fokker-Planck equation.

Findings

Bounded the dissipation of the distance to equilibrium in terms of the distance itself.

Derived new trend to equilibrium estimates in Wasserstein-2 like metric.

Applicable to cases with non-convex confinement potentials.

Abstract

A quadratic optimal transport metric on the set of probability measure over $\R^2$ is introduced. The quadratic cost is given by the euclidean norm on $\R^2$ associated to some well chosen symmetric positive matrix, which makes the metric equivalent to the usual Wasserstein-2 metric. The dissipation of the distance to the equilibrium along the kinetic Fokker-Planck flow, is bounded by below in terms of the distance itself. It enables to obtain some new type of trend to equilibrium estimate in Wasserstein-2 like metric, in the case of non-convex confinement potential.