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arXiv · 2607.03360

Smoluchowski-Kramers Approximation Meets Khasminskii Averaging Principles in Nonequilibrium Random Environments I

Hongjiang Qian
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Abstract

This work establishes a simultaneous Smoluchowski-Kramers approximation and Khasminskii averaging principle for a class of second-order stochastic differential equations (SDEs) in nonequilibrium random environments. The system describes the motion of a particle of mass $m>0$ subject to external forces, friction, and noise, all of which depend on a fluctuating environment such as a stochastic heat bath. The environment is modeled by a fast-varying first-order SDE, where a parameter $0<ε\ll 1$ encodes the time-scale separation. Under the scaling $m=ε^2$, the slow process converges in probability to an effective dynamics with averaged drift and noise-induced coefficients. Our analysis utilizes a pathwise integration-by-parts formula and Poisson equations associated with the fast dynamics. Finally, numerical experiments are provided for demonstration.

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Hongjiang Qian. 2026-07-03. Smoluchowski-Kramers Approximation Meets Khasminskii Averaging Principles in Nonequilibrium Random Environments I. https://arxiv.org/abs/2607.03360

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