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arXiv · 2609.29205

Smoluchowski-Kramers approximation with Lévy noise in the Meyer--Zheng topology

Xueru Liu·Qianqian Jiang·Wei Wang
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Abstract

We study the Smoluchowski-Kramers approximation for a stochastic wave equation with state-dependent damping on a bounded domain, driven by both a $Q$-Wiener process and a Lévy process with finite second moment. As $\varepsilon\to0$, we prove that $u^\varepsilon$ converges in distribution, in the Meyer-Zheng topology, to the unique weak solution of an overdamped stochastic parabolic equation. The proof relies on a nonlinear transformation associated with the damping coefficient, uniform energy estimates, and compactness arguments in the pseudo-path topology. We identify the limiting equation, which contains both the classical Gaussian noise-induced drift caused by state-dependent damping and an explicit jump correction generated by the Lévy noise. We show that the latter coincides exactly with the Marcus-to-Itô correction associated with the canonical jump flow induced by the nonlinear damping transformation.

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Xueru Liu, Qianqian Jiang, Wei Wang. 2026-09-24. Smoluchowski-Kramers approximation with Lévy noise in the Meyer--Zheng topology. https://arxiv.org/abs/2609.29205

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