arXiv subjects

Xue-Mei Li

Publications and source records attributed to Xue-Mei Li.

At least 19 recordsLinked to original sources

arXiv2026

Timelike Ollivier-Ricci curvature

We introduce a codimension one construction of coarse Ricci curvature in Lorentzian geometry. Using the $1$-Lorentz-Wasserstein distance $\ell_1$, we compare probability measures supported on small spacelike hypersurfaces through nearby events and recover, in a precise asymptotic regime, the ambient Ricci curvature in future-directed unit timelike directions, with a universal dimensional prefactor reflecting this codimension one construction. The construction echoes the Raychaudhuri equation, which relates the evolution of spatial volume expansion along timelike geodesics to Ricci curvature. Our quantitative estimates rely on transport maps with controlled displacement, constructed through a Moser-type flow. At leading order, the slice construction is insensitive to smooth weights. By smearing the spacelike slices into thin timelike tubes and calibrating the temporal and spatial scales, we recover the timelike Bakry-Émery tensor $\mathrm{Ric}+\mathrm{Hess}\,V$ associated with the weighted reference measure $\smash{\mathfrak{m}=\mathrm{e}^{-V}\,\mathrm{vol}_g}$.

math.DG↗

arXiv2026

Navier-Stokes with a fractional transport noise as a limit of multi-scale dynamics

We define a bona fide rough path solution for the Navier-Stokes equation with an additional rough transport term, and show that the SPDE on the three-dimensional torus driven by a fractional Brownian motion on $H^σ$ has solutions characterised as the effective limits of a slow/fast system. We further show that this rough path solution is equivalent to the widely used incremental notion of solution (the unbounded rough driver formulation), demonstrating broader applicability to other nonlinear SPDEs.

math.PR↗

arXiv2026

Strong completeness of SDEs and non-explosion for RDEs with coefficients having unbounded derivatives

We establish a non-explosion result for rough differential equations (RDEs) in which the noise and drift coefficients, together with their derivatives, may grow unboundedly at infinity. In addition, we prove the existence of a global bi-continuous solution flow for stochastic differential equations (SDEs). Finally, the non-explosion results for RDEs are shown to be sharp by constructing counterexamples.

math.PR↗

arXiv2025

Fluctuations from a random fractional averaging limit

We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the fractional averaging and fractional homogenization theorems of Hairer and Li (arXiv:1902.11251, arXiv:2109.06948), we establish a fluctuation result. The deviation of the slow motion, scaled by epsilon^{1/2-H}, from its effective, time-dependent random limit converges, as the time-separation scale epsilon tends to zero, to the solution of a stochastic differential equation driven by a fractional Brownian motion and influenced by an additional space--time Gaussian field. Since the averaging principle and the fractional homogenization hold in different modes of convergence, obtaining the required joint convergence is a delicate matter. Moreover, neither the continuity of the Ito--Lyons solution map nor the martingale method is directly applicable for our purposes, so the proof requires several innovations. To establish the fluctuation theorem, we combine cumulant methods with a residue lemma and formulate the enlarged system as a rough differential equation in a suitable space.

math.PR↗

arXiv2025

Stochastic nonlinear wave equation with rougher than white noise

We study the singular stochastic wave equation on $\mathbb T^2$, with a cubic nonlinearity and Gaussian rough Matérn forcing (a Fourier multiplier of order $α>0$ applied to space-time white noise) and establish local well-posedness for $α< \tfrac{3}{8}$. This extends [GKO18] beyond white noise and strengthens the quadratic-case result [OO21] ($α<\tfrac 12$). Our argument develops new trilinear estimates in Bourgain spaces together with sharp, case-specific cubic counting estimates.

math.PR↗

arXiv2025

A critical stochastic heat equation with long-range noise

We consider a semilinear stochastic heat equation in spatial dimension at least $3$, forced by a noise that is white in time with a covariance kernel that decays like $\lvert x\rvert^{-2}$ as $\lvert x\rvert\to\infty$. We show that in an appropriate diffusive scaling limit with a logarithmic attenuation of the noise, the pointwise statistics of the solution can be approximated by the solution to a forward-backward stochastic differential equation (FBSDE). The scaling and structure of the problem is similar to that of the two-dimensional stochastic heat equation forced by an approximation of space-time white noise considered by the first author and Gu (Ann. Probab., 2022). However the resulting FBSDE is different due to the long-range correlations of the noise.

math.PR↗

arXiv2025

Properties of the cores and filaments in the Ophiuchus molecular cloud and its L1688 hub-filament system

Analyzing filaments and cores in molecular clouds is key to understanding galactic star formation and its environmental dependence. This paper studies the properties and distribution of dense cores and filaments in the Ophiuchus molecular cloud, with a focus on the L1688 hub-filament system (HFS) and its star formation potential. We extracted sources and filaments from Herschel images and a 13.5 arcsec resolution surface density map using the getsf method, identified prestellar cores among the extracted sources, evaluated core mass segregation, and constructed the core mass function (CMF). We derived properties of the filaments from their radial surface density profiles, constructed the filament linear density function (FLDF), and assessed the mass distribution in the L1688 HFS to estimate the core and filament formation efficiencies (CFE, FFE). The results suggest that the filament-dominated core formation is a key mechanism in star formation within the system.

astro-ph.GA↗

arXiv2025

Coarse Ricci curvature of weighted Riemannian manifolds

We show that the generalized Ricci tensor of a weighted complete Riemannian manifold can be retrieved asymptotically from a scaled metric derivative of Wasserstein 1-distances between normalized weighted local volume measures. As an application, we demonstrate that the limiting coarse curvature of random geometric graphs sampled from Poisson point process with non-uniform intensity converges to the generalized Ricci tensor.

math.DG↗

arXiv2025

Coarse extrinsic curvature of Riemannian submanifolds

We introduce a novel concept of coarse extrinsic curvature for Riemannian submanifolds, inspired by Ollivier's notion of coarse Ricci curvature. This curvature is derived from the Wasserstein 1-distance between probability measures supported in the tubular neighborhood of a submanifold, providing new insights into the extrinsic curvature of isometrically embedded manifolds in Euclidean spaces. The framework also offers a method to approximate the mean curvature from statistical data, such as point clouds generated by a Poisson point process. This approach has potential applications in manifold learning and the study of metric embeddings, enabling the inference of geometric information from empirical data.

math.DG↗

arXiv2025

Fluctuations of stochastic PDEs with long-range correlations

We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-integrable spatial covariance that decays at the rate of $|x|^{-κ}$ at infinity, where $κ\in (2, d)$. Inspired by recent studies on SHE and KPZ equations driven by noise with compactly supported spatial correlation, we demonstrate that the correlations persist in the large-scale limit. The fluctuations of the diffusively scaled solution converge to the solution of a stochastic heat equation with additive noise whose correlation is the Riesz kernel of degree $-κ$. Moreover, the fluctuations converge as a distribution-valued process in the optimal Hölder topologies.

math.PR↗

arXiv2024

Scaling limit of the KPZ equation with non-integrable spatial correlations

We study the large scale fluctuations of the KPZ equation in dimensions $d \geq 3$ driven by Gaussian noise that is white in time Gaussian but features non-integrable spatial correlation with decay rate $κ\in (2, d)$ and a suitable limiting profile. We show that its scaling limit is described by the corresponding additive stochastic heat equation. In contrast to the case of compactly supported covariance, the noise in the stochastic heat equation retains spatial correlation with covariance $|x|^{-κ}$. Surprisingly, the noise driving the limiting equation turns out to be the scaling limit of the noise driving the KPZ equation so that, under a suitable coupling, one has convergence in probability, unlike in the case of integrable correlations where fluctuations are enhanced in the limit and convergence is necessarily weak.

math.PR↗

arXiv2023

On the (Non-)Stationary Density of Fractional-Driven Stochastic Differential Equations

We investigate the stationary measure $π$ of SDEs driven by additive fractional noise with any Hurst parameter and establish that $π$ admits a smooth Lebesgue density obeying both Gaussian-type lower and upper bounds. The proofs are based on a novel representation of the stationary density in terms of a Wiener-Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the non-stationary density, which extend previously known results in the additive setting. In addition, we study a parameter-dependent version of the SDE and prove smoothness of the stationary density, jointly in the parameter and the spatial coordinate. With this we revisit the fractional averaging principle of Li and Sieber [Ann. Appl. Probab. 32(5) (2022)] and remove an ad-hoc assumption on the limiting coefficients. Avoiding any use of Malliavin calculus in our arguments, we can prove our results under minimal regularity requirements.

math.PR↗

arXiv2023

Properties of the dense cores and filamentary structures in the Vela C molecular cloud

The initial and boundary conditions of the Galactic star formation in molecular clouds are not well understood. In an effort to shed new light on this long-standing problem, we measured properties of dense cores and filamentary structures in the Vela C molecular cloud, observed with Herschel. We applied the getsf extraction method to separate the components of sources and filaments from each other and their backgrounds, before detecting, measuring, and cataloging the structures. The cores and filamentary structures constitute 40% of the total mass of Vela C, most of the material is in the low-density molecular background cloud. We selected 570 reliable cores, of which 149 are the protostellar cores and 421 are the starless cores. Almost 78% of the starless cores were identified with the gravitationally bound prestellar cores. The exponent of the CMF (alpha = 1.35) is identical to that of the Salpeter IMF. We selected 68 filaments with at least one side that appeared not blended with adjacent structures. The filament widths are in the range of 0.15 pc to 0.63 pc, and have a median value of W = 0.3(0.11) pc. The surface densities of filaments are well correlated with their contrasts and linear densities. Within uncertainties of the filament instability criterion, many filaments may well be both supercritical and subcritical. A large fraction of filaments may definitely be considered supercritical, in which are found 94 prestellar cores, 83 protostellar cores, and only 1 unbound starless core. Taking into account the uncertainties, the supercritical filaments contain only prestellar and protostellar cores. Our findings support the idea that there exists a direct relationship between the CMF and IMF and that filaments play a key role in the formation of prestellar cores, which is consistent with the previous Herschel results.

astro-ph.GA↗

arXiv2022

Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $L^1$

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

math.PR↗

arXiv2022

Generating diffusions with fractional Brownian motion

We study fast / slow systems driven by a fractional Brownian motion $B$ with Hurst parameter $H\in (\frac 13, 1]$. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $Y^\varepsilon$ denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $\varepsilon \ll 1$, the solutions of the equation $$ dX^\varepsilon = \varepsilon^{\frac 12-H} F(X^\varepsilon,Y^\varepsilon)\,dB+F_0(X^\varepsilon,Y^\varepsilon)\,dt\; $$ converge to a regular diffusion without having to assume that $F$ averages to $0$, provided that $H< \frac 12$. For $H > \frac 12$, a similar result holds, but this time it does require $F$ to average to $0$. We also prove that the $n$-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuous interpolation between the homogenisation theorem for random ODEs with rapidly oscillating right-hand sides ($H=1$) and the averaging of diffusion processes ($H= \frac 12$).

math.PR↗

arXiv2022

Functional Limit Theorems for Volterra Processes and Applications to Homogenization

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process $(y_t)_{t\geq 0}$ in the rough path topology. As an application, we establish weak convergence as $\varepsilon\to 0$ of the solution of the random ordinary differential equation (ODE) $\frac{d}{dt}x^\varepsilon_t=\frac{1}{\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})$ and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type Itô SDE $dx_t=G(x_t,dt)$, where $G(x,t)$ is a semi-martingale with spatial parameters.

math.PR↗

arXiv2022

Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging

Our first result is a stochastic sewing lemma with quantitative estimates for mild incremental processes, with which we study SPDEs driven by fractional Brownian motions in a random environment. We obtain uniform $L^p$-bounds. Our second result is a fractional averaging principle admitting non-stationary fast environments. As an application, we prove a fractional averaging principle for SPDEs.

math.PR↗

arXiv2022

$Herschel$ investigation of cores and filamentary structures in the Perseus molecular cloud

Cores and filamentary structures are the prime birthplaces of stars, and play key roles in the process of star formation. Latest advances in the methods of multi-scale source and filament extraction, and in making high-resolution column density map from $Herschel$ multi-wavelength observations enable us to detect the filamentary network structures in highly complex molecular cloud environments. The statistics for physical parameters shows that core mass strongly correlates with core dust temperature, and $M/L$ strongly correlates with $M/T$, which is in line with the prediction of the blackbody radiation, and can be used to trace evolutionary sequence from unbound starless cores to robust prestellar cores. Crest column densities of the filamentary structures are clearly related with mass per unit length ($M_{\rm line}$), but are uncorrelated by three orders ranging from $\sim 10^{20}$ to $\sim 10^{22}$ $ \rm cm^{-2}$ with widths. Full width at half maximum (FWHM) have a median value of 0.15 pc, which is consistent with the 0.1 pc typical inner width of the filamentary structures reported by previous research. We find $\sim $70\% of robust prestellar cores (135/199) embedded in supercritical filaments with $M_{\rm line}>16~M_{\odot}/{\rm pc}$, which implies that the gravitationally bound cores come from fragmentation of supercritical filaments. And on the basis of observational evidences that probability distribution function (PDF) with power-law distribution in the Perseus south is flatter than north, YSO number is significantly less than that in the north, and dust temperature difference. We infer that south region is more gravitationally bound than north region.

astro-ph.GA↗