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.
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.
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 ...).
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.
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.
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 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 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.
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 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…
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.
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…
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.
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.
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 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.
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.
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.
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.
Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纪念大学格伦菲分校科学与环境学院）；Department of Mathematics, University of Connecticut（康涅狄格大学数学系）
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 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.
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…
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…
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…
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…
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…
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…
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 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…
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.
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 derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\dd)}$-transportation cost inequality $$W^\rr_{2/(2-\dd)}(fμ,μ)^{2/(2-\dd)}\le Cμ(f\log f), μ(f)=1, f\ge 0$$ for some constant $C>0$, and they are equivalent if the curvature of the corresponding generator is bounded below. Weighted log-Sobolev and entropy-cost inequalities are als…
An upper bound for the Wasserstein distance is provided in the general framework of the Wiener-Poisson space. Is obtained from this bound a second order Poincaré-type inequality which is useful in terms of computations. For completeness sake, is made a survey of these results on the Wiener space, the Poisson space, and the Wiener-Poisson space, and showed several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated field (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein-Uhlenbeck …
We improve the constant $\fracπ{2}$ in $L^1$-Poincaré inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\fracπ{2}}$. For Hamming cube the sharp constant is not known, and $\sqrt{\fracπ{2}}$ gives an estimate from below for this sharp constant. On the other hand, L. Ben Efraim and F. Lust-Piquard have shown an estimate from above: $C_1\le \fracπ{2}$. There are at least two other independent proofs of the same estimate from above (we write down them in this note). Since those proofs are very different from the proof of Ben Efraim and…
We prove some results about the super Poincaré inequality (SPI) and its relation to the spectrum of an operator: we show that it can be alternatively written with Orlicz norms instead of L1 norms, and we use this to give an alternative proof that a bound on the bottom of the essential spectrum implies a SPI. Finally, we apply these ideas to give a spectral proof of the log Sobolev inequality for the Gaussian measure.
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…
