Meyer-Zheng 拓扑下带 Lévy 噪声的 Smoluchowski-Kramers 近似
Smoluchowski-Kramers approximation with Lévy noise in the Meyer--Zheng topology
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中文总结 AI 辅助
研究带状态依赖阻尼和 Lévy 噪声的随机波动方程的 Smoluchowski-Kramers 近似，证明在 Meyer-Zheng 拓扑下收敛到过阻尼抛物方程，并识别出跳跃修正与 Marcus-to-Itô 修正一致。
中文摘要 AI 辅助
我们研究了有界域上由 $Q$-Wiener 过程和具有有限二阶矩的 Lévy 过程共同驱动的、带状态依赖阻尼的随机波动方程的 Smoluchowski-Kramers 近似。当 $\varepsilon\to0$ 时，我们证明了 $u^\varepsilon$ 在 Meyer-Zheng 拓扑下依分布收敛到过阻尼随机抛物方程的唯一次解。证明依赖于与阻尼系数相关的非线性变换、一致能量估计以及伪路径拓扑中的紧性论证。我们识别了极限方程，其中既包含由状态依赖阻尼引起的经典高斯噪声诱导漂移，也包含由 Lévy 噪声产生的显式跳跃修正。我们证明了后者与由非线性阻尼变换诱导的典范跳跃流相关的 Marcus-to-Itô 修正完全一致。
英文摘要
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.
发表机构
- Dalian University of Technology（大连理工大学）
- Ruhr University Bochum（波鸿鲁尔大学）
- Nanjing University（南京大学）
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