Optimal linear drift for the speed of convergence of an hypoelliptic diffusion

TL;DR

This paper demonstrates that among similar Ornstein-Uhlenbeck processes, a non-reversible hypoelliptic process achieves the fastest convergence to equilibrium, optimizing the speed of convergence for systems with the same invariant measure and randomness.

Contribution

It proves that a specific non-reversible hypoelliptic process maximizes the asymptotic convergence rate among all comparable Ornstein-Uhlenbeck processes.

Findings

Non-reversible hypoelliptic processes outperform reversible ones in convergence speed.

Optimal linear drift can be designed to maximize convergence rate.

The result applies to systems with identical invariant measures and noise inputs.

Abstract

Among all generalized Ornstein-Uhlenbeck processes which sample the same invariant measure and for which the same amount of randomness (a $N$-dimensional Brownian motion) is injected in the system, we prove that the asymptotic rate of convergence is maximized by a non-reversible hypoelliptic one.