We prove logarithmic Sobolev inequality for measures $$ q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb R^n, $$ under the assumptions that: (i) the conditional distributions $$ Q_i(\cdot| x_j, j\neq i)=\text{dist}(X_i| X_j= x_j, j\neq i) $$ satisfy a logarithmic Sobolev inequality with a common constant $ρ$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian $V$ are not too large relative to $ρ$. \bigskip Condition (ii) has the form that the norms of some matrices defined in terms of the mixed partial derivatives of $V$ do not exceed $1/2\cdotρ\cdot(1-\de)$. The logarithmic Sobolev constant of $q^n$ can then be estimated from below by $1/2\cdotρ\cdotδ$. This improves on earlier results by Th. Bodineau and B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant for $q^n$.
We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a box of diameter L may depend on L but not on the number of particles. This is a first, but relevant and quite technical step, in the proof that this logarithmic-Sobolev constant grows as L^2, that will be presented in a forthcoming paper.
In this paper, we prove modified logarithmic Sobolev inequalities for canonical ensembles with superquadratic single-site potential. These inequalities were introduced by Bobkov and Ledoux, and are closely related to concentration of measure and transport-entropy inequalities. Our method is an adaptation of the iterated two-scale approach that was developed by Menz and Otto to prove the usual logarithmic Sobolev inequality in this context. As a consequence, we obtain convergence in Wasserstein distance $W_p$ for Kawasaki dynamics on the Ginzburg-Landau model.
We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz'ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new…
We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on $\dR^n$, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality for all uniform strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.
Modified logarithmic Sobolev inequalities for Abelian quantum double models
TU München（慕尼黑工业大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；University of Cambridge（剑桥大学）；Universität Tübingen（蒂宾根大学）；Wuhan University（武汉大学）；Wuhan Institute of Quantum Technology（武汉量子技术研究院）；Politecnico di Milano（米兰理工大学）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Instituto de Ciencias Matemáticas（数学科学研究所）；Universidad Nacional de Educación a Distancia（西班牙国家开放大学）；Inria（法国国家信息与自动化研究所）；Institut Polytechnique de Paris（巴黎理工学院）
We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type $\exp(-|x|^\al)$ or more complex $\exp(-|x|^\al\log^β(2+|x|))$ ($\al\in]1,2[$ and $\be\in\dR$) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.
We study global existence, uniqueness and positivity of weak solutions of a class of reaction-diffusion systems of chemical kinetics type, under the assumptions of logarithmic Sobolev inequality and appropriate exponential integrability of the initial data.
In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.
The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of the topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one (Proposition 1.4), and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic So…
We develop in this paper an amelioration of the method given by S. Bobkov and M. Ledoux in GAFA (2000). We prove by Prekopa-Leindler Theorem an optimal modified logarithmic Sobolev inequality adapted for all log-concave measure on $\dR^n$. This inequality implies results proved by Bobkov and Ledoux, the Euclidean Logarithmic Sobolev inequality generalized in the last years and it also implies some convex logarithmic Sobolev inequalities for large entropy.
The de cit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances.
We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of Bobkov-Ledoux. Some consequences are also discussed.
In this note we give a simple, dimension independent, proof of the logarithmic Sobolev inequality on the Heisenberg groups $H_n=\R^{2n+1}$ using the measure preserving transformations of the Brownian motion. We have corrected some serious flaws and simplified the proofs.
We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.
We prove that the logarithmic Sobolev constant for the inhomogeneous symmetric nearest neighbour zero range process on a cube of size N^d grows as N^2. We apply this result to the inhomogeneous process which arises in the study of the homogeneous version of the zero range interacting particle system with colours.
Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincaré or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequality.
Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau [LPY] on spectral gap and logarithmic Sobolev inequalities for unbounded conservative spin systems, we study uniform bounds in these inequalities for Glauber dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n), (x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or small perturbation of strictly convex) and, following [LPY], the case V is a bounded perturbation of a quadratic potential. By a simple path counting argument for the standard random walk, uniform bounds for the Gl…
We identify sharp spaces and prove quantitative and non-quantitative stability results for the logarithmic Sobolev inequality involving Wasserstein and $L^p$ metrics. The techniques are based on optimal transport theory and Fourier analysis. We also discuss a probabilistic approach.
We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}(dγ)$ distance for $p>1$. To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner--Hirschman inequality.
We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic …
In this paper, we consider the measure determined by a fractional Ornstein-Uhlenbeck process. For such measure, we establish a martingale representation theorem and consequently obtain the Logarithmic-Sobolev inequality. To this end, we also present the integration by parts formula for such measure, which is obtained via its pull back formula and the Bismut method.
We are interested in the $q$ Logarithmic Sobolev inequality for probability measures on the infinite product of Heisenberg groups. We assume that the one site boundary free measure satisfies either a $q$ Log-Sobolev inequality or a U-Bound inequality, and we determine conditions so that the infinite dimensional Gibbs measure satisfies a $q$ Log-Sobolev inequality.
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 ...).
We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand's transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry-Emery curvature is not lower bounded. Several new examples are provided.
We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free improvement for a related result given in [Eldan '18].
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.
Logarithmic Sobolev inequalities are a fundamental class of inequalities that play an important role in information theory. They play a key role in establishing concentration inequalities and in obtaining quantitative estimates on the convergence to equilibrium of Markov processes. More recently, deep links have been established between logarithmic Sobolev inequalities and strong data processing inequalities. In this paper we study logarithmic Sobolev inequalities from a computational point of view. We describe a hierarchy of semidefinite programming relaxations which give certified lower boun…
We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application.
In this paper we study some applications of the Lévy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the Lévy-Ornstein-Uhlenbeck process.
This note presents a method based on Feynman-Kac semigroups for logarithmic Sobolev inequalities. It follows the recent work of Bonnefont and Joulin on intertwining relations for diffusion operators, formerly used for spectral gap inequalities, and related to perturbation techniques. In particular, it goes beyond the Bakry-{É}mery criterion and allows to investigate high-dimensional effects on the optimal logarithmic Sobolev constant.The method is illustrated on particular examples (namely Subbotin distributions and double-well potentials), for which explicit dimension-free bounds on the latte…
We provide explicit lower bounds for the deficit in the Gaussian logarithmic Sobolev inequality in terms of differential operators that are naturally associated with the so-called Stein characterization of the Gaussian distribution. The techniques are based on a crucial use of the representation of the relative Fisher information, along the Ornstein-Uhlenbeck semigroup, in terms of the Minimal Mean-Square Error from information theory.
We prove logarithmic Sobolev inequalities and concentration results for convex functions and a class of product random vectors. The results are used to derive tail and moment inequalities for chaos variables (in spirit of Talagrand and Arcones, Gine). We also show that the same proof may be used for chaoses generated by log-concave random variables, recovering results by Lochowski and present an application to exponential integrability of Rademacher chaos.
We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on $\mathbb{R}^n$ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of $L^p$-Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-…
We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.
The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $Γ$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schr{ö}dinger problem yields a variations interpretation…
We introduce a free version of the Stein kernel, relative to a semicircular law. We use it to obtain a free counterpart of the HSI inequality of Ledoux, Peccatti and Nourdin, which is an improvement of the free logarithmic Sobolev inequality of Biane and Speicher, as well as a rate of convergence in the (multivariate) entropic free Central Limit Theorem. We also compute the free Stein kernels for several relevant families of self-adjoint operators.
This note, mostly expository, is devoted to Poincar{é} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact …
We are interested in the Logarithmic Sobolev Inequality for the infinite volume Gibbs measure with no quadratic interactions. We consider unbounded spin systems on the one dimensional Lattice with interactions that go beyond the usual strict convexity and without uniform bound on the second derivative. We assume that the one dimensional single-site measure with boundaries satisfies the Log-Sobolev inequality uniformly on the boundary conditions and we determine conditions under which the Log-Sobolev Inequality can be extended to the infinite volume Gibbs measure.
In this work we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behaviour for the laws of solutions to stochastic differential equations.
We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a powerful logarithmic Sobolev inequality proved by L. Wu (2000), as well as on several variations of the so-called Herbst argument. We provide several applications, in particular to edge counting and more general length power functionals in random geometric graphs, as well as to the convex distance for random point measures recently introduced by M. Reitzner (2…
We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are …
In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the mo…
We show that for a hypoelliptic Dirichlet form operator A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A is not holomorphic.
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 th…
In this paper we establish quantitative results about the bridges of the Langevin dynamics and the associated reciprocal processes. They include an equivalence between gradient estimates for bridge semigroups and couplings, comparison principles, bounds of the distance between bridges of different Langevin dynamics, and a Logarithmic Sobolev inequality for bridge measures. The existence of an invariant measure for the bridges is also discussed and quantitative bounds for the convergence to the invariant measure are proved . All results are based on a seemingly new expression of the drift of a …
We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.
We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's $Γ-$ calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-kno…
We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.
We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that $Φ$ is a symmetric convex function on $\dR$ satisfying $(1+\e)Φ(x)\leq {x}Φ'(x)\leq(2-\e)Φ(x)$ for $x\geq0$ large enough and with $\e\in]0,1/2]$. We prove that the probability measure on $\dR$ $μ_Φ(dx)=e^{-Φ(x)}/Z_Φdx$ satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant $A,B,D>0$ such that for all smooth $f>0$, \begin{equation*} \ent{μ_Φ}{f^2}\leq A\int H_Φ\PAR{{\frac{f'}{f}}}f^2dμ_Φ, \text{with} H_Φ(x)= {\begin{array}{rl} Φ^*\PAR{Bx} &\text{if }\ABS{x}\geq D, x^2 …
Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576--1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017--1054]. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on $\{0,1\}^n$. This new bound is shown to be asymptotically optimal.
