# Phase Space Homogenization of Noisy Hamiltonian Systems

## TL;DR

This paper analyzes the limiting behavior of noisy Hamiltonian systems with inertia, deriving the homogenized joint distribution of position and velocity, and establishing convergence properties in the small mass limit.

## Contribution

It provides a rigorous derivation of the homogenized distribution for inertial particles under noise, including convergence rates, which advances understanding of stochastic Hamiltonian dynamics.

## Key findings

- Derived explicit expression for the limiting joint distribution.
- Proved weak convergence of the joint distributions.
- Established bounds on convergence rates for expected values.

## Abstract

We study the dynamics of an inertial particle coupled to forcing, dissipation, and noise in the small mass limit. We derive an expression for the limiting (homogenized) joint distribution of the position and (scaled) velocity degrees of freedom. In particular, weak convergence of the joint distributions is established, along with a bound on the convergence rate for a wide class of expected values.