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Xueru Liu

Publications and source records attributed to Xueru Liu.

7 recordsLinked to original sources
arXiv2026

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

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.

math.PR↗
arXiv2026

Stable and Gaussian Fluctuation Limit of a Lévy-Driven Slow-Fast System with Polynomial Dissipation

We study fluctuation limit of a slow-fast system driven by $α$-stable Lévy noise with $1<α<2.$ The slow component is generated by an odd polynomial function $f(y):=y^q,$ while in the fast component, the drift is $g(y):=-|y|^p\operatorname{sgn}(y)$ for some $p>0.$ Although the noise is given, the fluctuation limit is either a stable process or a Brownian motion, depending on both $p$ and $α.$ The critical line between stable limit and Brownian limit is $q+1-p=α/2.$

math.PR↗
arXiv2026

Approximation of Random Differential Equations Driven by Physical Brownian Motion with Fast Oscillating Noise

We investigate approximation of random differential equations driven by semimartingales satisfying a singularly perturbed Langevin equation with scaled mixing random force. By a diffusion approximation approach, we explore the limit of the rough path lift of this semimartingale, and a universal limit theorem is applied to identify the limit of random differential equation. A structurally parallel proof also applies to establish an iterated weak invariance principle for the mixing random force, which is itself an independent interesting result. We find that, the limit of both of the second-level processes, have the form of iterated integral of Stratonovich form plus an anti-symmetric part which is proportional to the time increment.

math.PR↗
arXiv2026

On the Approximation of Differential Equations Driven by Some Random Processes as Rough Paths

We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path theory. To this end, we lift the random process as a rough path in a natural manner. After suitable change-of-variable, the random process has a form of slow-fast system. Moment estimates of both the random process and its lift are given, followed by which, averaging technique and convergence theorem in rough path topology are used to identify the limit.

math.PR↗
arXiv2025

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

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.

math.PR↗
arXiv2024

The Smoluchowski-Kramers approximation for a system with arbitrary friction depending on both state and distribution

A system of stochastic differential equations describing diffusive phenomena, which has arbitrary friction depending on both state and distribution is investigated. The Smoluchowski-Kramers approximation is seen to describe dynamics in the small mass limit. We obtain the limiting equation and, in particular, the addition drift terms that appear in the limiting equation are expressed in terms of the solutions to the Lyapunov matrix equation and Sylvester matrix equation. Furthermore, we provide the rate of convergence and extend the system to encompass more general interactions and noise.

math.PR↗