Logarithmic Sobolev Research Articles

Corrigendum to “Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results”

This corrigendum corrects several mistakes in Appendix B.3 of [Ann. Inst. H. Poincaré Anal. Non Linéaire (2023). .

Hasimoto frames and the Gibbs measure of the periodic nonlinear Schrödinger equation

The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto’s theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.

Functional Inequalities for Some Generalised Mehler Semigroups

We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure \(\sigma \), we prove functional integral inequalities with respect to \(\sigma \), such as logarithmic Sobolev and Poincaré type. Consequently, some integrability properties of exponential functions with respect to \(\sigma \) are deduced.

Applications of p-harmonic transplantation for functional inequalities involving a Finsler norm

In this paper, we prove several inequalities such as Sobolev, Poincaré, logarithmic Sobolev, which involve a general norm with accurate information of extremals, and are valid for some symmetric functions. We use Ioku’s transformation, which is a special case of p-harmonic transplantation, between symmetric functions.

One-Dimensional Fokker–Planck Equations and Functional Inequalities for Heavy Tailed Densities

We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations modeling socio-economic problems, and one-dimensional functional inequalities of the type of Poincaré, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails. As main examples, we consider inequalities satisfied by inverse Gamma densities, taking values on mathbb R_+, and Cauchy-type densities, taking values on mathbb R.

Stability of the Bakry-Émery theorem on [formula omitted

We establish quantitative stability estimates for the Bakry-Émery bound on logarithmic Sobolev and Poincaré constants of uniformly log-concave measures. More specifically, we show that if a 1-uniformly log-concave measure has almost the same logarithmic Sobolev or Poincaré constant as the standard Gaussian measure, then it almost splits off a Gaussian factor. Our results are dimension-free, leading to dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs are based on Stein's method, optimal transport, and an approximate integration by parts identity relating measures and near-extremals in the associated functional inequality.

A family of Beckner inequalities under various curvature-dimension conditions

In this paper, we offer a proof for a family of functional inequalities interpolating between the Poincaré and the logarithmic Sobolev (standard and weighted) inequalities. The proofs rely both on entropy flows and on a CD(ρ,n) condition, either with ρ=0 and n>0, or with ρ>0 and n∈R. As such, results are valid in the case of a Riemannian manifold, which constitutes a generalization of what was previously proved.

A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions

For displacement convex functionals in the probability space equipped with the Monge–Kantorovich metric we prove the equivalence between the gradient and functional type Łojasiewicz inequalities. We also discuss the more general case of λ-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto–Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo–Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed.

Logarithmic Sobolev, isoperimetry and transport inequalities on graphs

Logarithmic Sobolev, isoperimetry and transport inequalities on graphs

Stein’s method, logarithmic Sobolev and transport inequalities

We develop connections between Stein’s approximation method, logarithmic Sobolev and transport inequalities by introducing a new class of functional inequalities involving the relative entropy, the Stein kernel, the relative Fisher information and the Wasserstein distance with respect to a given reference distribution on \({\mathbb{R}^{d}}\). For the Gaussian model, the results improve upon the classical logarithmic Sobolev inequality and the Talagrand quadratic transportation cost inequality. Further examples of illustrations include multidimensional gamma distributions, beta distributions, as well as families of log-concave densities. As a by-product, the new inequalities are shown to be relevant towards convergence to equilibrium, concentration inequalities and entropic convergence expressed in terms of the Stein kernel. The tools rely on semigroup interpolation and bounds, in particular by means of the iterated gradients of the Markov generator with invariant measure the distribution under consideration. In a second part, motivated by the recent investigation by Nourdin, Peccati and Swan on Wiener chaoses, we address the issue of entropic bounds on multidimensional functionals F with the Stein kernel via a set of data on F and its gradients rather than on the Fisher information of the density. A natural framework for this investigation is given by the Markov Triple structure (E, μ, Γ) in which abstract Malliavin-type arguments may be developed and extend the Wiener chaos setting.

Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2013 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments: finite and infinite-dimensional manifolds, metric spaces, fractals and graphs, including random environments. The workshop brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas. Unifying themes were heat kernel analysis, mass transport problems and related functional inequalities such as Poincar´e, Sobolev, logarithmic Sobolev, Bakry-Emery, Otto-Villani and Talagrand inequalities. These concepts were at the heart of Perelman’s proof of Poincar´e’s conjecture, as well as of the development of the Otto calculus, and the synthetic Ricci bounds of Lott-Sturm-Villani. The workshop provided participants with an opportunity to discuss how these techniques can be used to approach problems in optimal transport for non-local operators, subelliptic operators in finite and infinite dimensions, analysis on singular spaces, as well as random walks in random media.

Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

In this paper, consider the circular Cauchy distribution $\mu_x$ on the unit circle $S$ with index $0\le |x|<1$, we study the spectral gap and the optimal logarithmic Sobolev constant for $\mu_x$, denoted respectively as $\lambda_1(\mu_x)$ and $C_{\mathrm{LS}}(\mu_x).$ We prove that $\frac{1}{1+|x|}\le \lambda_1(\mu_x)\le 1$ while $C_{\mathrm{LS}}(\mu_x)$ behaves like $\log(1+\frac{1}{1-|x|})$ as $|x|\to 1.$

Cell polarisation model: The 1D case

We study the dynamics of a one-dimensional non-linear and non-local drift-diffusion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self-similar profile, to a steady state of finite time, blow-up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the sub-critical and critical mass cases. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.

Quantum logarithmic Sobolev inequalities and rapid mixing

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative \documentclass[12pt]{minimal}\begin{document}$\mathbb {L}_p$\end{document}Lp-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed. This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants. Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained, including the depolarizing semigroup and quantum expanders.

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincar\'e, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.

Extremal maps in Sobolev type inequalities: Some remarks

In this work we make some observations on the existence of extremal maps for sharp L 2 -Riemannian Sobolev type inequalities as Nash and logarithmic Sobolev ones. Among other results, we prove also that there exist smooth compact Riemannian manifolds with scalar curvature changing signal on which there exist extremal maps.

Some Applications of Mass Transport to Gaussian-Type Inequalities

As discovered by Brenier, mapping through a convex gradient gives the optimal transport in ℝ n . In the present article, this map is used in the setting of Gaussian-like measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.

Comment on: “Hypercontractivity of Hamilton–Jacobi equations”, by S. Bobkov, I. Gentil and M. Ledoux

In their remarkable work [1], Bobkov, Gentil and Ledoux improve, generalize and simplify most previously known results about hypercontractivity, logarithmic Sobolev and transport inequalities. Among other results, they are able to recover and generalize the HWI inequalities introduced in [4], (1) DV ≥ KId ⇒ H(f |e−V ) ≤ W (f, e−V ) √ I(f |e−V )− K 2 W (f, e−V ). Here f and g are arbitrary probability distribution functions, H(f |g) ≡ ∫ f log(f/g) is the Kullback relative entropy of f w.r.t g, I(f |g) ≡ ∫ f |∇ log(f/g)|2 is the relative Fisher information of f w.r.t g, and

Sobolev type inequalities for general symmetric forms

A general version of the Sobolev type inequality, including both the classical Sobolev inequality and the logarithmic Sobolev one, is studied for general symmetric forms by using isoperimetric constants. Some necessary and sufficient conditions are presented as results. The main results are illustrated by two examples of birth-death processes.

Interpolation between logarithmic Sobolev and Poincare inequalities

This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincare inequalities. For such generalized Poincare inequalities we improve upon the known constants from the literature. Cette note est consacree a des inegalites intermediaires qui interpolent entre les inegalites de Sobolev logarithmiques et les inegalites de Poincare. Pour de telles inegalites de Poincare generalisees, nous ameliorons les constantes donnees dans la litterature.