Periodic Homogenization for Hypoelliptic Diffusions

TL;DR

This paper investigates the long-term behavior of a hypoelliptic diffusion process with periodic drift, demonstrating convergence to Brownian motion and providing bounds on the convergence rate.

Contribution

It introduces a homogenization framework for hypoelliptic diffusions with periodic coefficients, deriving explicit covariance and convergence rates.

Findings

Law of the process converges to Brownian motion under diffusive scaling

Explicit expression for the limiting covariance matrix

Provides upper bounds on the convergence rate

Abstract

We study the long time behavior of an Ornstein-Uhlenbeck process under the influence of a periodic drift. We prove that, under the standard diffusive rescaling, the law of the particle position converges weakly to the law of a Brownian motion whose covariance can be expressed in terms of the solution of a Poisson equation. We also derive upper bounds on the convergence rate.