Nathael Gozlan, Cyril Roberto, Paul-Marie Samson
We prove that a probability measure on an abstract metric space satisfies a non trivial dimension free concentration inequality for the $\ell_2$ metric if and only if it satisfies the Poincaré inequality.
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A note on the equivalence of super-Poincaré inequality
本文研究超皮卡德不等式与弱log-Sobolev不等式及超log-Sobolev不等式之间的等价性，并建立相关速率函数的显式关系。
A sharp Poincaré-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The new inequality is nothing but an infinitesimal form of Ehrhard's inequality for the Gaussian measure.
We prove that for symmetric Markov processes of diffusion type admitting a "carré du champ", the Poincaré inequality is equivalent to the exponential convergence of the associated semi-group in one (resp. all) $Ł^p(μ)$ spaces for $p\in (1,+\infty)$. Part of this result extends to the stationary non necessarily symmetric situation.
The aim of the paper is to study the pinned Wiener measure on the loop space over a simply connected compact Riemannian manifold together with the Hilbert space structure induced by Mallianvin calculus and the induced Ornstein- Uhlenbeck operator. We give a concrete estimate for the weak Poincare inequality for the O-U Dirichlet form on loop space over simply connected compact Riemannian manifold with strict positive Ricci curvature.
We prove that for a probability measure on $\mathbb{R}^n$, the Poincaré inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan et al. and Feldheim et al., concerning probability measures on the real line. The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest. We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by …
We consider probability measures supported on a finite discrete interval $[0,n]$. We introduce a new finitedifference operator $\nabla_n$, defined as a linear combination of left and right finite differences. We show that this operator $\nabla_n$ plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures.
We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian $d\times d$ matrix valued functions, with the same Poincaré constant. This generalizes recent results [Aoun et al. 2019, Kathuria 2019] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.
In this paper, we prove that, in dimension one, the Poincaré inequality is equivalent to a new transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the chi-square pseudo-distance. We also check tensorization of this transport-chi-square inequality.
Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
本文基于马尔可夫半群与Γ-微分算子，推导加权高斯测度下的广义贝克纳不等式及皮卡德不等式稳定性，进一步研究其梯度和L^2稳定性，并应用至海森堡不确定性原理分析及对数Sobolev不等式构建。
The priority of Liming Wu for the Poincaré inequality for birth-death processes on a Poisson space is announced.
In this short note we give a refinement of Brascamp-Lieb in the style of Houdre-Kagan extension for Poincare inequality in one dimension. This is inspired by works of Helffer and Ledoux.
We present two extensions of the one dimensional free Poincaré inequality similar in spirit to two classical refinements.
The paper is a continuation of our paper [12,2], and it studies functional inequalities for non-local Dirichlet forms with finite range jumps or large jumps. Let $α\in(0,2)$ and $μ_V(dx)=C_Ve^{-V(x)}\,dx$ be a probability measure. We present explicit and sharp criteria for the Poincaré inequality and the super Poincaré inequality of the following non-local Dirichlet form with finite range jump $$\mathscr{E}_{α, V}(f,f):= (1/2)\iint_{|x-y|\le 1}\frac{(f(x)-f(y))^2}{|x-y|^{d+α}} dy μ_V(dx);$$ on the other hand, we give sharp criteria for the Poincaré inequality of the non-local Dirichlet form wi…
Stability of the Poincaré-Korn inequality
本文在矩约束下解决了 Carrapatoso 等人关于 Poincaré-Korn 不等式锐常数高斯最优性的问题，并证明接近最优常数的测度在定量意义上接近标准高斯分布。
In this paper, we establish the super Poincaré inequality for the two-parameter Dirichlet process when the partition number of the state space is finite. Furthermore, if the partition number is infinite, the super Poincar'e inequality doesn't hold. To overcome the difficulty caused by the degenerency of the diffusion coefficient on the boundary of the domain, localization method and perturbation argument in [14] are effective.
We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...).
On Gilles Pisier's approach to gaussian concentration, isoperimetry and Poincaré-type inequalities
本研究拓展了Gilles Pisier的研究方法，可在欧氏空间旋转不变概率分布（含多维柯西测度）中探索高斯测度相关集中、等周与庞加莱型不等式结果的对应形式。
By using Lyapunov conditions, weak Poincaré inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit results we study concrete examples.
The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincare inequality but not the log-Sobolev one.
We show that the quadratic transportation cost inequality $T_2$ is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmic-Sobolev inequalities by Bobkov and Ledoux \cite{BL97}, that are shown to hold as soon as a Poincaré inequality is satisfied.
We characterize the symmetric measures which satisfy the one dimensional convex Poincaré inequality. For these measures the tenesorization argument yields concentration inequalities for their products and convex sets in R^n.
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in \cite{Bau2}. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is …
Convergence of non-reversible Markov processes via lifting and flow Poincar{é} inequality
本文提出基于二阶提升和亚强制性变分方法的通用框架，引入流庞加莱不等式与一般散度引理，用于非可逆马尔可夫过程的定量收敛分析，并成功应用于非平衡统计力学模型和带一般随机跳跃核的分段确定性采样过程。
We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the $L^2$ variance of an admissible function is controlled by the homogeneous $H^1$ norm. In the case of Loop spaces, it was observed by L. Gross that the homogeneous $H^1$ norm alone may not control the $L^2$ norm and a potential term involving the end value of the Brownian bridge is introduced. Aida, on the other hand, introduced a weight on the Dirichlet form. We show that Aida's modified Logarithmic Sobolev inequality implies weak Logarithmic Sobolev Ine…
This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument.
This paper is concerned with a class of singular stable-like Dirichlet forms on $\R^d$, which are generated by $d$ independent copies of a one-dimensional symmetric $α$-stable process, and whose Lévy jump kernel measure is concentrated on the union of the coordinate axes. Explicit and sharp criteria for Poincaré inequality, super Poincaré inequality and weak Poincaré inequality of such singular Dirichlet forms are presented. When the reference measure is a product measure on $\R^d$, we also consider the entropy inequality for the associated Dirichlet forms, which is similar to the log-Sobolev …
We show sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d \in \mathbb{N}$. Here we focus on differentiable functions on the Euclidean space in presence of a Poincaré-type inequality. The bounds are based on $d$-th order derivatives.
We extend a Gaussian functional inequality to a countable product of Gaussian measures. This inequality improves on the classical Poincare inequality for Gaussian measures. As an application, we prove that First Passage Percolation has sublinear variance when the edge times distribution belongs to a wide class of continuous distributions, including the exponential one. This extends a result by Benjamini, Kalai and Schramm, valid for positive Bernoulli edge times.
We prove a deviation inequality for noncommutative martingales by extending Oliveira's argument for random matrices. By integration we obtain a Burkholder type inequality with satisfactory constant. Using continuous time, we establish noncommutative Poincaré type inequalities for "nice" semigroups with a positive curvature condition. These results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to Bobkov and Götze in the commutative case. To demonstrate our setting is general enough, we give various examples, including certain group von Neuma…
Sub-Gaussian heat kernel estimates are typical of fractal graphs. We show that sub-Gaussian estimates on graphs follow from a Poincaré inequality, capacity upper bound, and a slow volume growth condition. An important feature of this work is that we do not assume elliptic Harnack inequality, cutoff Sobolev inequality, or exit time bounds.
We present an improved version of the second order Gaussian Poincaré inequality, firstly introduced in Chatterjee (2009) and Nourdin, Peccati and Reinert (2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative CLTs for non-linear functionals of stationary Gaussian fields related to the Breuer-Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.
We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong…
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger's inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincaré inequalities and …
We provide a new exponential concentration inequality for First Passage Percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm which gave a variance bound for Bernoulli edge times. Our approach is based on some functional inequalities extending the work of Rossignol and Falik and Samorodnitsky.
We study Talagrand concentration and Poincaré type inequalities for unbounded pure jump Markov processes. In particular we focus on processes with degenerate jumps that depend on the past of the whole system, based on the model introduced by Galves and Löcherbach in \cite{G-L}, in order to describe the activity of a biological neural network. As a result we obtain exponential rates of convergence to equilibrium.
In this paper, we consider Poincaré inequalities for non euclidean metrics on $\mathbb{R}^d$. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincaré type inequalities in terms of transportation-cost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized Beckner-Latala-Oleszkiewicz ine…
We consider Markov random fields of discrete spins on the lattice $\Zd$. We use a technique of coupling of conditional distributions. If under the coupling the disagreement cluster is "sufficiently" subcritical, then we prove the Poincaré inequality. In the whole subcritical regime, we have a weak Poincaré inequality and corresponding polynomial upper bound for the relaxation of the associated Glauber dynamics.
We generalize the Beckner's type Poincaré inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $μ\starν$, where $μ$ is the reference Gaussian measure and $ν$ is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and in…
It is known that the Poincaré inequality is equivalent to the quadratic transportation-variance inequality (namely $W_2^2(fμ,μ) \leqslant C_V \mathrm{Var}_μ(f)$), see Jourdain \cite{Jourdain} and most recently Ledoux \cite{Ledoux18}. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_V$ than before, which equals the double of Poincaré constant. Applying the same argument leads to more characterizations of the Poincaré inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow…
This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincaré inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.
In this paper, the Harnack inequalities and super-Poincaré inequality for generalized CIR model are obtained. Since the noise is degenerate, the intrinsic metric has been introduced to construct the coupling by change of measure. By using isopermetric constant, the optimal estimate of the rate function in the super Poincaré inequality for the associated Dirichlet form is also obtained.
Let $X$ be a regular linear continuous positively recurrent Markov process with state space $\R$, scale function $S$ and speed measure $m$. For $a\in \R$ denote B^+_a&=\sup_{x\geq a} \m(]x,+\infty[)(S(x)-S(a)) B^-_a&=\sup_{x\leq a} \m(]-\infty;x[)(S(a)-S(x)) We study some characteristic relations between $B^+_a$, $B^-_a$, the exponential moments of the hitting times $T_a$ of $X$, the Hardy and Poincaré inequalities for the Dirichlet form associated with $X$. As a corollary, we establish the equivalence between the existence of exponential moments of the hitting times and the spectral gap of th…
For any $N\ge 2$ and $å:=(å_1,\cdots, å_{N+1})\in (0,\infty)^{N+1}$, let $μ^{(N)}_å$ be the corresponding Dirichlet distribution on $\DD:= \big\{ x=(x_i)_{1\le i\le N}\in [0,1]^N:\ \sum_{1\le i\le N} x_i\le 1\big\}.$ We prove the Poincaré inequality $$μ^{(N)}_å(f^2)\le \ff 1 {å_{N+1}} \int_{\DD}\Big\{\Big(1-\sum_{1\le i\le N} x_i\Big) \sum_{n=1}^N x_n(\pp_n f)^2\Big\}μ^{(N)}_å(\d x)+μ^{(N)}_å(f)^2,\ f\in C^1(\DD)$$ and show that the constant $\ff 1 {å_{N+1}}$ is sharp. Consequently, the associated diffusion process on $\DD$ converges to $μ^{(N)}_å$ in $L^2(μ^{(N)}_å)$ at the exponentially rate…
We prove a concentration inequality which asserts that, under some mild regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits pseudorandom behavior.
Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in \cite{BG}, then the following properties hold: 1 The volume doubling property; 2 The Poincaré inequality; 3 The parabolic Harnack inequality. The key ingredient is the study of dimensional reverse log-Sobolev inequalities for the heat semi…
This paper is concerned with the problem of nonlinear filter stability of ergodic Markov processes. The main contribution is the conditional Poincaré inequality (PI), which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE). Based on these dual formalisms, a comparison is drawn between the stochastic stability of a Markov process and the filter stability. The latter relies on the conditional PI describ…
Inspired by the idea of stochastic quantization proposed by Parisi and Wu, we construct the transition probability matrix which plays a central role in the renormalization group through a stochastic differential equation. By establishing the discrete time stochastic dynamics, the renormalization procedure can be characterized from the perspective of probability. Hence, we will focus on the investigation of the infinite dimensional stochastic dynamic. From the stochastic point of view, the discrete time stochastic dynamic can induce a Markov chain. Via calculating the square field operator and …
The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider $\varepsilon$-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of $\mathbb{R}^K$. We show that a volume doubling condition (VD) and local Poincaré inequality (LPI) hold for the random geometric graph (with high probability, and uniformly over all shortest path distance balls in a certain radius range) under suitable regularity conditions of the underlying submanifold a…
This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincaré inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincaré inequality, as well as upper and lower bounds for non-symmetric heat kernels.
Low-Temperature Asymptotics of the Poincaré and the log-Sobolev Constants for Łojasiewicz Potentials
研究低温下Poincaré不等式常数和log-Sobolev常数的行为，推翻Chewi和Stromme的猜想，确定一维情况的正确渐进行为。
Weak Poincaré inequality comparisons for ideal and hybrid slice sampling
本文利用弱庞加莱不等式框架比较混合与理想切片抽样的狄利克雷型，证明二者在适当条件下可相互继承快速收敛性，并应用于分析多种切片抽样算法的收敛行为。
We establish a Shearer-type inequality for the Poincaré constant, showing that the Poincaré constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincaré constants corresponding to convolutions of subsets of measures. This implies, for example, that the Poincaré constant is non-increasing along the central limit theorem. We also establish a dimension-free stability estimate for subadditivity of the Poincaré constant on convolutions which uniformly improves an earlier one-dimensional estimate of a similar nature by Johnson (2004). As a bypr…
By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincaré and weak Poincaré inequalities are established, while another Poincaré type inequality is disproved under some reasonable assumptions.
A connection between the semigroup of the Cauchy process killed upon exiting a domain $D$ and a mixed boundary value problem for the Laplacian in one dimension higher known as the "mixed Steklov problem," was established in a previous paper of the authors. From this, a variational characterization for the eigenvalues $λ_n$, $n\geq 1$, of the Cauchy process in $D$ was obtained. In this paper we obtain a variational characterization of the difference between $λ_n$ and $λ_1$. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound esti…
In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdös-Rényi random graph are discussed. The number of vertices of fixed degree is also studie…
In this note, we obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas in [8].
We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.
On the discrete Poincaré inequality for B-schemes of 1D Fokker-Planck equations in full space
本文针对全空间一维 Fokker-Planck 方程的 B-格式，提出基于 Gamma 微积分和 Lyapunov 函数方法的两种途径推导离散 Poincaré 不等式，并由此证明离散格式向平衡态的指数收敛。
In in this paper we establish an explicit and sharp estimate of the spectral gap (Poincaré inequality) and the transportation inequality for Gibbs measures, under the Dobrushin uniqueness condition. Moreover, we give a generalization of the Liggett's $M-ε$ theorem for interacting particle systems.
Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincaré--Lions inequalities
该研究针对弱约束下不满足庞加莱不等式的欠阻尼朗之万动力学，基于加权庞加莱不等式适配胖尾/次指数势能场景，借助新的时空加权庞加莱-利昂斯不等式，给出了L^∞初始数据在L^2范数下的显式收敛速率估计。
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