Long term convergence rate of Smoluchowski-Kramers approximation by Stein's method

TL;DR

This paper investigates the convergence rate of the Smoluchowski-Kramers approximation for stochastic differential equations using Stein's method, establishing explicit bounds on the Wasserstein distance between invariant measures as mass tends to zero.

Contribution

It provides the first explicit convergence rate bounds for the invariant distributions in the Smoluchowski-Kramers approximation using Stein's method.

Findings

Wasserstein distance between invariant measures is of order O(√m|ln m|).

In 1D, the convergence rate improves to O(√m).

The results quantify the long-term behavior convergence as mass approaches zero.

Abstract

We consider the following second-order stochastic differential equation on $\mathbb{R}^{2d}$: \begin{equation*} dX_t^m=Y_t^mdt, \quad mdY_t^m=b(X_t^m)dt+\sigma(X_t^m)dB_t-Y^m_tdt, \end{equation*} where $X^m_t$ and $Y^m_t$ represent the position and velocity of a particle at time $t$, $m>0$ denotes its mass, $b:\mathbb{R}^d \rightarrow \mathbb{R}^d$ is the drift field, $\sigma:\mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$ is the diffusion coefficient, and $\{B_t\}_{t \ge 0}$ is a $d$-dimensional standard Brownian motion. The Smoluchowski--Kramers approximation states that as $m \rightarrow 0$, this system converges to the limiting equation: \begin{equation*} dX_t=b(X_t)dt+\sigma(X_t)dB_t. \end{equation*} Utilizing Stein's method, we prove that the $1$-Wasserstein distance between the invariant distribution of $X_t^m$ and that of its small-mass limit $X_t$ is of order $O(\sqrt{m}|\ln m|).$ Particularly, in the one-dimensional case, the convergence rate can be improved to $O(\sqrt{m}).$