On the small mass limit of stochastic wave equation driven by cylindrical stable process

TL;DR

This work analyzes the small mass limit for a stochastic wave equation driven by cylindrical $\alpha$-stable noise ($\alpha\in(1,2)$) and shows convergence to a stochastic heat equation as the mass parameter $\varepsilon\to0$. A three-part velocity splitting technique enables detailed moment estimates, while a tightness argument based on the infinite-dimensional Aldous condition yields convergence in the Skorokhod space. The authors establish well-posedness for both the stochastic wave and heat equations in this non-Gaussian setting and obtain a rate of convergence $\mathbb{E}\sup_{t\le T}\|U^{\varepsilon}(t)-\bar{U}^{\varepsilon}(t)\|_{-2}\lesssim\varepsilon^{\theta}$ when $\theta>0$. The results extend known SK-type approximations to cylindrical stable noise and provide a rigorous route from wave-type dynamics to diffusive, heat-type behavior in infinite dimensions, with implications for singular perturbations and SPDE modeling under heavy-tailed noise.

Abstract

We explore the small mass limit of a stochastic wave equation (SWE) driven by cylindrical $α$-stable noise, where $α\in (1,2)$, and prove that it converges to a stochastic heat equation. We establish its well-posedness, and in particular, the càdlàg property, which is not trivial in the infinite dimensional case. Using a splitting technique, we decompose the velocity component into three parts, which gives convenience to the moment estimate. We show the tightness of solution of SWE by verifying the infinite dimensional version of Aldous condition. After these preparation, we pass the limit and derive the approximation equation.

Theorems & Definitions (23)

• Remark 2.1
• Theorem 2.2
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Proposition 3.4
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• proof