Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium

TL;DR

This paper derives an optimal non-symmetric Fokker-Planck equation with linear drift in d6d for fastest exponential convergence to a prescribed anisotropic Gaussian equilibrium, explicitly constructing it with hypocoercive properties.

Contribution

It introduces a method to explicitly construct a non-symmetric Fokker-Planck equation with maximal decay rate and minimal multiplicative constant for a given Gaussian equilibrium.

Findings

Maximum decay rate equals the largest eigenvalue of the inverse covariance matrix.

Constructed Fokker-Planck equation has a diffusion matrix of rank one, ensuring hypocoercivity.

Achieved decay estimates with constants arbitrarily close to the infimum of 1.

Abstract

This paper is concerned with finding Fokker-Planck equations in $\mathbb{R}^d$ with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum. Such an optimal Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. In an $L^2$-analysis, we find that the maximum decay rate equals the maximum eigenvalue of the inverse covariance matrix, and that the infimum of the attainable multiplicative constant is 1, corresponding to the high-rotational limit in the Fokker-Planck drift.