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Liming Wu

Publications and source records attributed to Liming Wu.

At least 37 records · Page 2Linked to original sources
arXiv2020

Long-time behaviors of mean-field interacting particle systems related to McKean-Vlasov equations

In this paper, we investigate gradient estimate of the Poisson equation and the exponential convergence in the Wasserstein metric $W_{1,d_{l^1}}$, uniform in the number of particles, and uniform-in-time propagation of chaos for the mean-field weakly interacting particle system related to McKean-Vlasov equation. By means of the known approximate componentwise reflection coupling and with the help of some new cost function, we obtain explicit estimates for those three problems, avoiding the technical conditions in the known results. Our results apply when the confinement potential $V$ has many wells, the interaction potential $W$ has bounded second mixed derivative $\nabla^2_{xy}W$ which should be not too big so that there is no phase transition. As an application, we obtain the concentration inequality of the mean-field interacting particle system with explicit and sharp constants, uniform in time. Several examples are provided to illustrate the results.

math.PR↗
arXiv2019

Poincar{é} and logarithmic sobolev inequalities for nearly radial measures

If Poincar{é} inequality has been studied by Bobkov for radial measures, few is known about the logarithmic Sobolev inequalty in the radial case. We try to fill this gap here using different methods: Bobkov's argument and super-Poincar{é} inequalities, direct approach via L1-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee-Vempala in the logconcave bounded case are refined for radial measures.

math.FA↗
arXiv2019

The Kinetic Fokker-planck Equation With Mean Field Interaction

We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 ($μ$) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

math.AP↗
arXiv2019

Uniform Poincar{é} and logarithmic Sobolev inequalities for mean field particles systems

In this paper we establish some explicit and sharp estimates of the spectral gap and the log-Sobolev constant for mean field particles system, uniform in the number of particles, when the confinement potential have many local minimums. Our uniform log-Sobolev inequality, based on Zegarlinski's theorem for Gibbs measures, allows us to obtain the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate constant, generalizing the result of [10] by means of the displacement convexity approach, or [19, 20] by Bakry-Emery technique or the recent [9] by dissipation of the Wasserstein distance.

math.PR↗
arXiv2019

Ricci curvature and $W_1$-exponential convergence of Markov processes on graphs

In this paper, we show that the Ricci curvature lower bound in Ollivier's Wasserstein metric sense of a continuous time jumping Markov process on a graph can be characterized by some optimal coupling generator and provide the construction of this latter. Some previous results of Ollivier for discrete time Markov chains are generalized to the actual continuous time case. We propose a comparison technique with some death-birth process on $\mathbb N$ to obtain some explicit exponential convergence rate, by modifying the metric. A counterpart of Zhong-Yang's estimate is established in the case where the Ricci curvature with repsect to the graph metric is nonnegative. Moreover we show that the Lyapunov function method for the exponential convergence works with some explicit quantitative estimates, once if the Ricci curvature is bounded from below by a negative constant. Finally we present applications to Glauder dynamics under some dynamical versions of the Dobrushin uniqueness condition or of the Dobrushin-Shlosman analyticity condition.

math.PR↗
arXiv2019

Large deviations for empirical measures of mean field Gibbs measures

In this paper, we show that the empirical measure of mean-field model satisfies the large deviation principle with respect to the weak convergence topology or the stronger Wasserstein metric, under the strong exponential integrability condition on the negative part of the interaction potentials. In contrast to the known results we prove this without any continuity or boundedness condition on the interaction potentials. The proof relies mainly on the law of large numbers and the exponential decoupling inequality of de la Pena for U-statistics.

math.PR↗
arXiv2017

Centered Sobolev inequality and exponential convergence in $Φ$-entropy

In this short paper we find that the Sobolev inequality $$\frac 1{p-2}\left[\left(\int f^{p} dμ\right)^{2/p} - \int f^2 dμ\right] \le C \int |\nabla f|^2 dμ$$ ($p\ge 0$) is equivalent to the exponential convergence of the Markov diffusion semigroup $(P_t)$ to the invariant measure $μ$, in some $Φ$-entropy. We provide the estimate of the exponential convergence in total variation and a bounded perturbation result under the Sobolev inequality. Finally in the one-dimensional case we get some two-sided estimates of the Sobolev constant by means of the generalized Hardy inequality.

math.PR↗
arXiv2017

Exponential convergence in the Wasserstein metric $W_1$ for one dimensional diffusions

In this paper, we find some general and efficient sufficient conditions for the exponential convergence $W_{1,d}(P_t(x,\cdot), P_t(y,\cdot) )\le Ke^{-δt}d(x,y)$ for the semigroup $(P_t)$ of one-dimensional diffusion. Moreover some sharp estimates of the involved constants $K\ge 1, δ>0$ are provided. Those general results are illustrated by a series of examples.

math.PR↗
arXiv2017

First-principles studies of the Optical anisotropy of R3 space group chalcogenides crystal AX2MQ6

In order to explain the reason that all the sythesised \ce{AX2MQ6} chalcogenides compounds are phase-matching while isomorphic chalcogenides compounds \ce{AX4M5Q12} are not phase-matched materials. The linear optical property birefringence and non-linear optical property of $R3$ space group crystals \ce{AX2MQ6} have been calculated to compare with \ce{AX4M5Q12}. Their Electron Localization Function(ELF) was calculated to show that the acentricity of the bonds on $sp^3$ hybridization \ce{Q^2-} ions is the main origin of optical anisotropy. To quantificat the acentricity, a geometric parameter dihedral angle between tetrahedral undersides and xy-planes was defined. We find theoretically the birefringence depend on the defined geometric parameter and ion radius. This relation between birefringence and structure resonalbly explain the abnormal difference of birefringence of isostructural \ce{AX2MQ6} and \ce{AX4M5Q12} and useful for exploring new phase-matching IR-NLO crystals.

cond-mat.mtrl-sci↗
arXiv2015

Log-Sobolev, isoperimetry and transport inequalities on graphs

In this paper, we study some functional inequalities (such as Poincaré inequalities, logarithmic Sobolev inequalities, generalized Cheeger isoperimetric inequalities, transportation-information inequalities and transportation-entropy inequalities) for reversible nearest-neighbor Markov processes on a connected finite graph by means of (random) path method. We provide estimates of the involved constants.

math.PR↗
arXiv2014

Convergence rate and concentration inequalities for Gibbs sampling in high dimension

The objective of this paper is to study the Gibbs sampling for computing the mean of observable in very high dimension - a powerful Markov chain Monte Carlo method. Under the Dobrushin's uniqueness condition, we establish some explicit and sharp estimate of the exponential convergence rate and prove some Gaussian concentration inequalities for the empirical mean.

math.ST↗
arXiv2013

$L^1$-Uniqueness of the Fokker-Planck equation on a Riemannian manifold

In this paper, we obtain a necessary and sufficient condition for $L^{\infty}$-uniqueness of Sturm-Liouville operator $a(x)\frac{d^2}{dx^2} + b(x) \frac d{dx} -V$ on an open interval of $\rr$, which is equivalent to the $L^1$-uniqueness of the associated Fokker-Planck equation. For a general elliptic operator $\LL^V:=Δ+b \cdot\nabla -V$ on a Riemannian manifold, we obtain sharp sufficient conditions for the $L^1$-uniqueness of the Fokker-Planck equation associated with $\LL^V$, via comparison with a one-dimensional Sturm-Liouville operator. Furthermore the $L^1$-Liouville property is derived as a direct consequence of the $L^\infty$-uniqueness of $\LL^V$.

math.CA↗
arXiv2012

Moderate Deviations for Mean Field Particle Models

This article is concerned with moderate deviation principles of a general class of mean eld type interacting particle models. We discuss functional moderate deviations of the occupation measures for both the strong -topology on the space of fi nite and bounded measures as well as for the corresponding stochastic processes on some class of functions equipped with the uniform topology. Our approach is based on an original semigroup analysis combined with stochastic perturbation techniques and projective limit large deviation methods.

math.PR↗
arXiv2011

On the concentration properties of Interacting particle processes

These lecture notes present some new concentration inequalities for Feynman-Kac particle processes. We analyze different types of stochastic particle models, including particle profile occupation measures, genealogical tree based evolution models, particle free energies, as well as backward Markov chain particle models. We illustrate these results with a series of topics related to computational physics and biology, stochastic optimization, signal processing and bayesian statistics, and many other probabilistic machine learning algorithms. Special emphasis is given to the stochastic modeling and the quantitative performance analysis of a series of advanced Monte Carlo methods, including particle filters, genetic type island models, Markov bridge models, interacting particle Markov chain Monte Carlo methodologies.

math.NA↗
arXiv2011

Transportation inequalities: From Poisson to Gibbs measures

We establish an optimal transportation inequality for the Poisson measure on the configuration space. Furthermore, under the Dobrushin uniqueness condition, we obtain a sharp transportation inequality for the Gibbs measure on $\mathbb{N}^Λ$ or the continuum Gibbs measure on the configuration space.

math.ST↗
arXiv2010

Bernstein type's concentration inequalities for symmetric Markov processes

Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.

math.PR↗
arXiv2009

Transportation-information inequalities for Markov processes (II) : relations with other functional inequalities

We continue our investigation on the transportation-information inequalities $W_pI$ for a symmetric markov process, introduced and studied in \cite{GLWY}. We prove that $W_pI$ implies the usual transportation inequalities $W_pH$, then the corresponding concentration inequalities for the invariant measure $μ$. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion process implies $W_1I$ (a result due to \cite{GLWY}) and a Cheeger type's isoperimetric inequality. Finally we exhibit relations between transportation-information inequalities and a family of functional inequalities (such as $Φ$-log Sobolev or $Φ$-Sobolev).

math.PR↗