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PreprintCC BY 4.0arXiv

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 ε→0, we prove that uε 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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Cite

BibTeXRIS
BibTeX
@misc{liu2026smoluchowskikramers,
  title = {{Smoluchowski-Kramers approximation with Lévy noise in the Meyer--Zheng topology}},
  author = {Xueru Liu and Qianqian Jiang and Wei Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29205},
  eprint = {2609.29205},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29205},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Smoluchowski-Kramers approximation with Lévy noise in the Meyer--Zheng topology
AU  - Xueru Liu
AU  - Qianqian Jiang
AU  - Wei Wang
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29205
UR  - https://arxiv.org/abs/2609.29205
AB  - 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 ε→0, we prove that u^ε 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.
ER  -