Central limit theorem for the Smoluchowski--Kramers approximation
Abstract
We establish the central limit theorem (CLT) for Smoluchowski--Kramers approximation of Langevin equations with state-dependent, possibly non-symmetric friction and multiplicative noise. To the best of our knowledge, this is the first such result, even in the case of constant friction and additive noise. The result differs in an essential way from classical averaging CLTs and reveals a new two-scale fluctuation phenomenon. The normalized difference process retains a persistent fast component and is generally not tight in C([0,T];ℝd), thus the usual continuous-path CLT cannot hold. We derive an exact decomposition of the normalized difference into an accumulated slow fluctuation and a rapidly relaxing velocity correction. The auxiliary slow fluctuation converges to a linear diffusion driven by both the original Brownian motion and an additional independent Brownian noise whose effective covariance is characterized explicitly through a quadratic Poisson corrector. For every fixed t>0, the fast correction converges conditionally to its local Gaussian equilibrium. Consequently, the fixed-time fluctuation limit is identified as a Gaussian mixture: (X̅t−Xtε)/√ε Longrightarrow Y̅t+N1/2(X̅t)ξ, where X̅t is the overdamped limit, Y̅t is the limiting slow fluctuation, N is the covariance of the terminal velocity correction and solves a Lyapunov equation, and ξ is a standard Gaussian random variable independent of (X̅t,Y̅t). A quantitative convergence rate is also provided. The result fills the remaining gap in the asymptotic theory of the overdamped approximation at the central limit scale, complementing the existing law of large numbers, moderate deviation and large deviation results.
Related
- Wasserstein bounds for invariant measures in the Smoluchowski-Kramers approximation with stable noise
- Smoluchowski-Kramers approximation with Lévy noise in the Meyer--Zheng topology
- A Functional Central Limit Theorem For A Rank-Killed Particle System With Persistent Copies
- A central limit theorem for the degree of a random product of Cremona transformations
- A Central Limit Theorem for Coprime Pair Counts in Randomly Translated Disks
BibTeX
@misc{li2026central,
title = {{Central limit theorem for the Smoluchowski--Kramers approximation}},
author = {Yun Li and Longjie Xie and Li Yang and Qianning Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30835},
eprint = {2610.30835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30835},
note = {CC0},
}
RIS
TY - UNPB
TI - Central limit theorem for the Smoluchowski--Kramers approximation
AU - Yun Li
AU - Longjie Xie
AU - Li Yang
AU - Qianning Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30835
UR - https://arxiv.org/abs/2610.30835
AB - We establish the central limit theorem (CLT) for Smoluchowski--Kramers approximation of Langevin equations with state-dependent, possibly non-symmetric friction and multiplicative noise. To the best of our knowledge, this is the first such result, even in the case of constant friction and additive noise. The result differs in an essential way from classical averaging CLTs and reveals a new two-scale fluctuation phenomenon. The normalized difference process retains a persistent fast component and is generally not tight in C([0,T];ℝ^d), thus the usual continuous-path CLT cannot hold. We derive an exact decomposition of the normalized difference into an accumulated slow fluctuation and a rapidly relaxing velocity correction. The auxiliary slow fluctuation converges to a linear diffusion driven by both the original Brownian motion and an additional independent Brownian noise whose effective covariance is characterized explicitly through a quadratic Poisson corrector. For every fixed t>0, the fast correction converges conditionally to its local Gaussian equilibrium. Consequently, the fixed-time fluctuation limit is identified as a Gaussian mixture: (X̅_t−X_t^ε)/√ε Longrightarrow Y̅_t+N^1/2(X̅_t)ξ, where X̅_t is the overdamped limit, Y̅_t is the limiting slow fluctuation, N is the covariance of the terminal velocity correction and solves a Lyapunov equation, and ξ is a standard Gaussian random variable independent of (X̅_t,Y̅_t). A quantitative convergence rate is also provided. The result fills the remaining gap in the asymptotic theory of the overdamped approximation at the central limit scale, complementing the existing law of large numbers, moderate deviation and large deviation results.
ER -