arXiv subjects

Huaiqian Li

Publications and source records attributed to Huaiqian Li.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic Formulas for Negative Sobolev Norms and Applications

This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power $p$ converges to a dimension-independent constant multiple of the corresponding $L^p$ norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian--Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.

math.PR↗

arXiv2025

On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting

We investigate the limiting behavior of Besov seminorms and nonlocal perimeters in Dunkl theory. The present work generalizes two fundamental results: the Maz'ya--Shaposhnikova formula for Gagliardo seminorms and the asymptotics of (relative) fractional $s$-perimeters. Our main contributions are twofold. First, we establish a dimension-free Maz'ya--Shaposhnikova formula via a novel, robust approach that avoids reliance on the density property of Besov spaces, offering broader applicability. Second, we prove limit formulas for nonlocal perimeters relative to bounded open sets $Ω$, removing boundary regularity assumptions in the forward direction, while introducing a weakened regularity condition on $\partialΩ$ (admitting fractal boundaries) for the converse, a significant improvement over existing requirements. To the best of our knowledge, the results in this second part are new even in the classic Laplacian setting.

math.FA↗

arXiv2025

Semi-log-convexity of ${\rm M}/{\rm M}/\infty$ queues on $\mathbb{Z}_+$

We solve the problem left in the recent paper by N. Gozlan et al [Potential Analysis 58, 2023, 123--158], establishing the semi-log-convexity of semigroups associated with ${\rm M}/{\rm M}/\infty$ queuing processes on the set of non-negative integers. Our approach is global in nature and yields the sharp constant.

math.PR↗

arXiv2025

Littlewood--Paley--Stein Square Functions for the Fractional Discrete Laplacian on $\mathbb{Z}$

We investigate the boundedness of ``vertical'' Littlewood--Paley--Stein square functions for the nonlocal fractional discrete Laplacian on the lattice $\mathbb{Z}$, where the underlying graphs are not locally finite. When $q\in[2,\infty)$, we prove the $l^q$ boundedness of the square function by exploring the corresponding Markov jump process and applying the martingale inequality. When $q\in (1,2]$, we consider a modified version of the square function and prove its $l^q$ boundedness through a careful in on the generalized carré du champ operator. A counterexample is constructed to show that it is necessary to consider the modified version. Moreover, we extend the study to a class of nonlocal Schrödinger operators for $q\in (1,2]$.

math.PR↗

arXiv2025

Exponential Ergodicity in $\W_1$ for SDEs with Distribution Dependent Noise and Partially Dissipative Drifts

Being concerned with ergodicity of McKean--Vlasov SDEs, we establish a general result on exponential ergodicity in the $L^1$-Wasserstein distance. The result is successfully applied to non-degenerate and multiplicative Brownian motion cases, degenerate second order systems, and even the additive $α$-stable noise, where the coefficients before the noise are allowed to be distribution dependent and the drifts are only assumed to be partially dissipative. Our results considerably improve existing ones whose coefficients before the noise are distribution-free.

math.PR↗

arXiv2023

Asymptotics of Kantorovich Distance for Empirical Measures of the Laguerre Model

We estimate the rate of convergence for the Kantorovich (or Wasserstein) distance between empirical measures of i.i.d. random variables associated with the Laguerre model of order $α$ on $(0,\infty)^N$ and their common law, which is not compactly supported and has no rotational symmetry. Compared with the Gaussian case, our result is sharp provided the parameter $α$ and the dimension $N$ are chosen in a specified regime.

math.PR↗

arXiv2023

Wasserstein Convergence for Empirical Measures of Subordinated Fractional Brownian Motions on the Flat Torus

We estimate rates of convergence for empirical measures associated with the subordinated fractional Brownian motion to the uniform distribution on the flat torus under the Wasserstein distance $\mathbb{W}_p$ for all $p\geq1$. In particular, our results coincides with recent ones on the diffusion process and the fractional Brownian motion. As an application, we provide similar results for time-discretized subordinated fractional Brownian motions.

math.PR↗

arXiv2023

Sharp Li--Yau Inequalities for Dunkl Harmonic Oscillators

We study the Li--Yau inequality for the heat equation corresponding to the Dunkl harmonic oscillator, which is a non-local Schrödinger operator parameterized by reflections and multiplicity functions. In the particular case when the reflection group is isomorphic to $\mathbb{Z}_2^d$, the result is sharp in the sense that equality is achieved by the heat kernel of the classic harmonic oscillator. We also provide the application on parabolic Harnack inequalities.

math.AP↗

arXiv2022

Wasserstein Convergence Rates for Empirical Measures of Subordinated Processes on Noncompact Manifolds

The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than [arXiv:2107.11568], generalize the recent ones in [Stoch. Proc. Appl. 144(2022), 271--287] and are shown to be sharp by a typical example. The proof is motivated by the aforementioned works.

math.PR↗

arXiv2022

Wasserstein Convergence for Empirical Measures of Subordinated Dirichlet Diffusions on Riemannian Manifolds

We investigate long-time behaviors of empirical measures associated with subordinated Dirichlet diffusion processes on a compact Riemannian manifold $M$ with boundary $\partial M$ to some reference measure, under the quadratic Wasserstein distance. For any initial distribution not concentrated on $\partial M$, we obtain the rate of convergence and even the precise limit for the conditional expectation of the quadratic Wasserstein distance conditioned on the process killed upon exiting $M\setminus\partial M$. In particular, the results coincide with the recent ones proved by F.-Y. Wang in \cite{eW2} for Dirichlet diffusion processes.

math.PR↗

arXiv2022

Wasserstein Convergence for Conditional Empirical Measures of Subordinated Dirichlet Diffusions on Riemannian Manifolds

The asymptotic behaviour of empirical measures has plenty of studies. However, the research on conditional empirical measures is limited. Being the development of Wang \cite{eW1}, under the quadratic Wasserstein distance, we investigate the rate of convergence of conditional empirical measures associated to subordinated Dirichlet diffusion processes on a connected compact Riemannian manifold with absorbing boundary. We give the sharp rate of convergence for any initial distribution and prove the precise limit for a large class of initial distributions. We follow the basic idea of Wang, but allow ourselves substantial deviations in the proof to overcome difficulties in our non-local setting.

math.PR↗

arXiv2021

Weak Type Estimates for Square Functions of Dunkl Heat Flows

The weak $(1,1)$ boundedness of the Littlewood--Paley--Stein square function for the Dunkl heat flow is proved via estimates on the Dunkl heat kernel of integral type and the Caldrón--Zygmund decomposition, which is the continuity of the recently work [arXiv:2003.11843] where the dimension-free $L^p$ boundedness of the same square function is studied.

math.CA↗

arXiv2021

Dimension-Free Square Function Estimates for Dunkl Operators

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for Dunkl heat flows in $\mathbb{R}^d$ is introduced by employing the full "gradient" induced by the corresponding carré du champ operator and then the $L^p$ boundedness is studied for all $p\in(1,\infty)$. For $p\in(1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in[2,\infty)$, we restrict to a particular case when the corresponding Weyl group is isomorphic to $\mathbb{Z}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry--Emery, which may be of independent interest, plays a crucial role. The results are dimension-free.

math.PR↗

arXiv2021

Li--Yau Inequalities for Dunkl Heat Equations

Motivated by recent works due to Yu--Zhao [J. Geom. Anal. 2020] and Weber--Zacher [arXiv:2012.12974], we study Li--Yau inequalities for the heat equation corresponding to the Dunkl Laplacian, which is a non-local operator parameterized by reflection groups and multiplicity functions. The results are sharp in the particular case when the reflection group is isometric to $\mathbb{Z}_2^d$.

math.AP↗

arXiv2021

Weak $(1,1)$ Boundedness of Riesz Transforms on Vector Bundles

The weak $(1,1)$ boundedness of (local) Riesz transforms corresponding to a large class of Schrödinger operators on vector bundles is proved, mainly assuming the generalized volume doubling condition, either Gaussian or sub-Gaussian upper bounds for the heat kernel only in short time, and derivative estimates of Bismut type for the corresponding semigroup.

math.PR↗

arXiv2019

Weighted Littlewood--Paley inequalities for heat flows in $\RCD$ spaces

We establish inequalities on vertical Littlewood--Paley square functions for heat flows in the weighted $L^2$ space over metric measure spaces satisfying the $\RCD^\ast(0,N)$ condition with $N\in [1,\infty)$ and the maximum volume growth assumption. In the noncompact setting, the later assumption can be removed by showing that the volume of the ball growths at least linearly. The estimates are sharp on the growth of the 2-heat weight and the 2-Muckenhoupt weight considered. The $p$-Muckenhoupt weight and the $p$-heat weight are also compared for all $p\in(1,\infty)$.

math.PR↗

arXiv2019

Littlewood--Paley--Stein inequalities on $\textup{RCD}(K,\infty)$ spaces

The $L^p$ boundedness on vertical Littlewood--Paley square functions for heat flows on $\textup{RCD}(K,\infty)$ spaces with $K\in\mathbb{R}$ is proved. With regards to the proof, for $1<p\leq 2$, Stein's analytical method is applied, while for $2<p<\infty$, the probabilistic approach in the sense of Bañuelos--Bogdan--Luks introduced recently is employed.

math.PR↗

arXiv2018

Littlewood--Paley--Stein Estimates for Non-local Dirichlet Forms

We obtain the boundedness in $L^p$ spaces for all $1<p<\infty$ of the so-called vertical Littlewood--Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For $1<p\le 2$, the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the finite jumping kernel case to the general case, while for $2\le p<\infty$, the Burkholder--Davis--Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those ones for pure jump symmetric Lévy processes in Euclidean spaces.

math.PR↗