arXiv subjects

Liping Xu

Publications and source records attributed to Liping Xu.

12 recordsLinked to original sources

arXiv2026

Quantitative propagation of chaos for the Boltzmann equation with moderately soft potentials

We study the Kac particle system associated with the spatially homogeneous Boltzmann equation with non-cutoff collision kernels in the moderately soft potential regime $-1<\gamma<0$. We prove uniqueness in law for the particle system and, in particular, establish an explicit convergence rate from the empirical measure of the Kac particle system to the weak solution of the Boltzmann equation in Wasserstein-2 distance, assuming that the initial datum $f_0$ has finite Fisher information and polynomial moments. To the best of our knowledge, this provides the first quantitative convergence rate for the Kac particle system in the soft potential setting.

math.AP↗

arXiv2026

Strichartz estimates for orthonormal systems on compact manifolds: the non-sharp region

We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the non-sharp admissible region of exponents, covering wave, Klein-Gordon, and fractional Schr\"odinger equations. Our approach combines the result of Wang-Zhang-Zhang \cite{wang2025strichartz} on the sharp admissible line with a Lieb-Sobolev inequality derived from a recent Cwikel estimate due to Sukochev-Yang-Zanin \cite{sukochev2025singular}, along with an alternative globalization method based on localized weak Lorentz estimates. Our results extend the Euclidean results of Bez-Hong-Lee-Nakamura-Sawano \cite{bez2019strichartz} and Bez-Lee-Nakamura \cite{bez2021strichartz}, as well as the classical single-function estimates on manifolds due to Kapitanski \cite{kapitanski1989some}, Burq-G\'erard-Tzvetkov \cite{MR2058384}, and Dinh \cite{dinh2016strichartz}.

math.CA↗

arXiv2026

Central limit theorem for a partially observed interacting system of Hawkes processes I: subcritical case

We consider a system of $N$ Hawkes processes and observe the actions of a subpopulation of size $K \le N$ up to time $t$, where $K$ is large. The influence relationships between each pair of individuals are modeled by i.i.d.Bernoulli($p$) random variables, where $p \in [0,1]$ is an unknown parameter. Each individual acts at a {\it baseline} rate $\mu > 0$ and, additionally, at an {\it excitation} rate of the form $N^{-1} \sum_{j=1}^{N} \theta_{ij} \int_{0}^{t} \phi(t-s)\,dZ_s^{j,N}$, which depends on the past actions of all individuals that influence it, scaled by $N^{-1}$ (i.e. the mean-field type), with the influence of older actions discounted through a memory kernel $\phi \colon \mathbb{R}{+} \to \mathbb{R}{+}$. Here, $\mu$ and $\phi$ are treated as nuisance parameters. The aim of this paper is to establish a central limit theorem for the estimator of $p$ proposed in \cite{D}, under the subcritical condition $\Lambda p < 1$.

math.PR↗

arXiv2025

Scaling limit for supercritical nearly unstable Hawkes processes with heavy tail

In this paper, we establish the asymptotic behavior of {\it supercritical} nearly unstable Hawkes processes with a power law kernel. We find that, the Hawkes process in our context admits a similar equation to that in \cite{MR3563196} for {\it subcritical} case. In particular, the rescaled Hawkes process $(Z^n_{nt}/n^{2\alpha})_{t\in[0,1]}$ converges in law to a kind of integrated fractional Cox Ingersoll Ross process with different coefficients from that in \cite{MR3563196}, as $n$ tends to infinity.

math.PR↗

arXiv2024

Rate of convergence of the Kac particle system for the Boltzmann equation with hard potentials

In this paper, we prove that the Kac stochastic particle system converges to the weak solution of the spatially homogeneous Boltzmann equation for hard potentials and hard spheres. We give, under the initial data with finite exponential moment assumption, an explicit rate of propagation of chaos in squared Wasserstein distance with quadratic cost by using a double coupling technique.

math.PR↗

arXiv2024

Scaling limits for supercritical nearly unstable Hawkes processes

In this paper, we investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than one and close to one as time goes to infinity. We find that,the scaling size determines the scaling behavior of the processes like in \cite{MR3313750}. Specifically,after suitable rescaling, the limit of the sequence of Hawkes processes is deterministic.And also with another appropriate rescaling, the sequence converges in law to an integrated Cox Ingersoll Ross like process. This theoretical result may apply to model the recent COVID19 in epidemiology and in social network.

math.PR↗

arXiv2020

Rate of convergence of the Kac-like particle system

In this paper, we consider the Kac stochastic particle system associated to the spatially homogeneous Boltzmann equation for true hard potentials. We establish a rate of propagation of chaos of the particle system to the unique solution of the Boltzmann equation. We use a probabilistic coupling method and give, under suitable assumptions on the initial condition, a rate of convergence of the empirical measure of the particle system to the solution of the Boltzmann equation for this singular interaction.

math.PR↗

arXiv2017

Evidence for short-range magnetic order in the nematic phase of FeSe from anisotropic in-plane magnetostriction and susceptibility measurements

The nature of the nematic state in FeSe remains one of the major unsolved mysteries in Fe- based superconductors. Both spin and orbital physics have been invoked to explain the origin of this phase. Here we present experimental evidence for frustrated, short-range magnetic order, as suggested by several recent theoretical works, in the nematic state of FeSe. We use a combination of magnetostriction, susceptibility and resistivity measurements to probe the in-plane anisotropies of the nematic state and its associated fluctuations. Despite the absence of long-range magnetic order in FeSe, we observe a sizable in-plane magnetic susceptibility anisotropy, which is responsible for the field-induced in-plane distortion inferred from magnetostriction measurements. Further we demonstrate that all three anisotropies in FeSe are very similar to those of BaFe2As2, which strongly suggests that the nematic phase in FeSe is also of magnetic origin.

cond-mat.supr-con↗

arXiv2016

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

We study some jumping SDE and the corresponding Fokker-Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_t)_{t\in [0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_t)_{t\in[0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli [5] concerning continuous SDEs and local PDEs.

math.PR↗

arXiv2016

Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials

We prove a strong/weak stability estimate for the 3D homogeneous Boltzmann equation with moderately soft potentials ($γ\in(-1,0)$) using the Wasserstein distance with quadratic cost. This in particular implies the uniqueness in the class of all weak solutions, assuming only that the initial condition has a finite entropy and a finite moment of sufficiently high order. We also consider the Nanbu $N$-stochastic particle system which approximates the weak solution. We use a probabilistic coupling method and give, under suitable assumptions on the initial condition, a rate of convergence of the empirical measure of the particle system to the solution of the Boltzmann equation for this singular interaction.

math.PR↗

arXiv2015

The Multifractal Nature of Boltzmann Processes

We consider the spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We study the pathwise properties of the stochastic process $(V_t)_{t\geq 0}$, which describes the time evolution of the velocity of a typical particle. We show that this process is almost surely multifractal and we compute its spectrum of singularities. For hard potentials, we also compute the multifractal spectrum of the position process $(X_t)_{t\ge0}$.

math.PR↗

arXiv2014

Path Relinking for Bandwidth Coloring Problem

A Path Relinking algorithm is proposed for the Bandwidth Coloring problem and the Bandwidth MultiColoring problem. It combines a population based relinking method and a tabu search based local search procedure. The proposed algorithm is assessed on two sets of 66 benchmark instances commonly used in the literature. Computational results demonstrate that the proposed algorithm is highly competitive in terms of both solution quality and efficiency compared to the best performing algorithms in the literature. Specifically, it improves the previous best known results for 15 out of 66 instances, while matching the previous best known results for 47 cases. Some key elements of the proposed algorithm are investigated.

cs.DM↗