Long term convergence rate of Smoluchowski-Kramers approximation by Stein's method
Shiyu Liu, Wei Liu, Lihu Xu
TL;DR
This work quantifies the long-time convergence rate of the Smoluchowski--Kramers approximation for small mass $m$ using Stein's method. By solving a Stein equation for the limiting diffusion and establishing strong ergodicity results for both the kinetic system and its limit, the authors derive explicit $1$-Wasserstein bounds between the invariant measures: $W_1(\nu_m, \nu) \le C \sqrt{m} |\log m|$ for $d>1$, and an improved $O(\sqrt{m})$ rate in 1D under additional smoothness. The key ingredients are moment bounds, a careful decomposition of the Stein error into tractable terms, and sharp regularity estimates for the Stein solution, including Hölder and log-Lipschitz controls. The results provide a rigorous, quantitative bridge between the kinetic system and its small-mass limit with potential extensions to other stochastic perturbations and higher-dimensional settings.
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+σ(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, $σ:\mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$ 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+σ(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 $O(\sqrt{m}).$