Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion

TL;DR

This paper investigates how to optimally modify linear drift in non-reversible diffusions to maximize convergence speed to equilibrium, providing theoretical solutions and practical algorithms.

Contribution

It introduces a method to find optimal non-reversible perturbations for linear diffusions that enhance convergence rates, with proven existence and implementable algorithms.

Findings

Optimal perturbations exist for linear diffusions.

The proposed algorithm effectively constructs these perturbations.

Numerical experiments confirm improved convergence rates.

Abstract

We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.