Logarithmic Sobolev Inequality Research Articles

Some functional inequalities on non-reversible Finsler manifolds

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by Ohta and Sturm. Following the approach of the $$\Gamma $$ -calculus of Bakry et al (2014), we show the dimensional versions of the Poincare–Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti and Mondino (2015) in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.

A discrete log-Sobolev inequality under a Bakry–Émery type condition

We consider probability mass functions $V$ supported on the positive integers using arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry--\'{E}mery condition for a Markov birth and death operator with invariant measure $V$. Under this condition, we prove a modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.

Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs

Abstract Studying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE ′ ⁢ ( n , 0 ) {\mathrm{CDE}^{\prime}(n,0)} , which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying CDE ′ ⁢ ( n , 0 ) {\mathrm{CDE}^{\prime}(n,0)} ) also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. As a consequence, we obtain that the dimension of the space of harmonic functions on graphs with polynomial growth is finite. In the Riemannian setting, this was originally a conjecture of Yau, which was proved in that context by Colding and Minicozzi. Under the assumption that a graph has positive curvature, we derive a Bonnet–Myers-type theorem. That is, we show the diameter of positively curved graphs is finite and bounded above in terms of the positive curvature. This is accomplished by proving some logarithmic Sobolev inequalities.

Existence of the global solution for fractional logarithmic Schrödinger equation

In this paper we consider the initial boundary value problem for a class of fractional logarithmic Schrödinger equation. By using the fractional logarithmic Sobolev inequality and introducing a family of potential wells, we give some properties of the family of potential wells and obtain existence of global solution.

Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups

We show that for a hypoelliptic Dirichlet form operator A A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A A is not holomorphic.

Coarse Ricci Curvature as a Function on $$\varvec{M\times M}$$

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula $$\begin{aligned} \mathrm {Ric}(\gamma ^{\prime }\left( 0\right) ,\gamma ^{\prime }\left( 0\right) )=\frac{1}{2}\frac{{d}^{2}}{{d}s^{2}}\mathrm {Ric}_{\triangle _{g} }(x,\gamma \left( s\right) ) \end{aligned}$$ for any curve $$\gamma (s)$$ .

Family of Functional Inequalities for the Uniform Measure

We consider on the interval [-1,1] the heat semigroup generated by the Legendre operator acting on the Hilbert space with respect to the uniform measure By means of a simple method involving some semigroup techniques, we describe a large family of optimal integral inequalities with the Poincaré and logarithmic Sobolev inequalities as particular cases.

Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains

Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincaré inequality. Their proof is based on the Bakry–Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli–Laplace models, and random transposition models, and to a finite-volume discretization of a one-dimensional Fokker–Planck equation, applying results by Mielke.

Weighted Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities

Motivated by recent results on the best constants and extremal functions for a family of the Caffarelli–Kohn–Nirenberg inequalities in [17, 23], we will study weighted Moser–Onofri–Beckner inequalities on the Euclidean space $$ \mathbb {R} ^{N}$$ . We also set up sharp weighted versions of the logarithmic Sobolev inequalities together with their best constants and optimizers.

Free Stein kernels and an improvement of the free logarithmic Sobolev inequality

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.

Volume Growth of Shrinking Gradient Ricci-Harmonic Soliton

In this paper, we study the shrinking gradient Ricci-harmonic soliton. Firstly using Chow–Lu–Yang’s argument, we give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci-harmonic solitons with $$S\ge \delta $$ to have polynomial volume growth with order $$n-2\delta $$ . Secondly, we derive a Logarithmic Sobolev inequality, as an application, we prove that any noncompact shrinking gradient Ricci-harmonic soliton must have linear volume growth, generalizing previous result of Munteanu and Wang (Commun Anal Geom 20(1):55–94, 2012).

Asymptotic Behavior for a Class of Logarithmic Wave Equations with Linear Damping

We consider the initial boundary value problem for a class of logarithmic wave equations with linear damping. By constructing a potential well and using the logarithmic Sobolev inequality, we prove that, if the solution lies in the unstable set or the initial energy is negative, the solution will grow as an exponential function in the $$H^1_0(\Omega )$$ norm as time goes to infinity. If the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of the energy. These results are extensions of earlier results.

Global Solution and Blow-up for a Class of p-Laplacian Evolution Equations with Logarithmic Nonlinearity

The main goal of this work is to study an initial boundary value problem for a quasilinear parabolic equation with logarithmic source term. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions.

A Stein deficit for the logarithmic Sobolev inequality

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.

Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

We consider a Fokker–Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear.In this paper we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy F on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an implicit Euler scheme obtained by minimizing hF(ϱ)+W22(ϱ0,ϱ) among all ϱ satisfying the constraint for some ϱ0 and time-step h>0.Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities.

Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity

In this paper, we study the semilinear elliptic equations with the sign-changing logarithmic nonlinearity. Under some conditions we obtain at least two nontrivial solutions by Nehari manifold and logarithmic Sobolev inequality. Here the results are quite different from these in the polynomial case.

Global solution and blow-up for a class of pseudo [formula omitted]-Laplacian evolution equations with logarithmic nonlinearity

The main goal of this work is to study an initial–boundary value problem for a nonlinear pseudoparabolic equation with logarithmic nonlinearity. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow-up of weak solutions.

A stroll along the gamma

We provide the first in-depth study of the Laguerre interpolation scheme between an arbitrary probability measure and the gamma distribution. We propose new explicit representations for the Laguerre semigroup as well as a new intertwining relation. We use these results to prove a local De Bruijn identity which hold under minimal conditions. We obtain a new proof of the logarithmic Sobolev inequality for the gamma law with α≥1/2 as well as a new type of HSI inequality linking relative entropy, Stein discrepancy and standardized Fisher information for the gamma law with α≥1/2.

Logarithmic Sobolev inequality revisited

We provide a new characterization of the logarithmic Sobolev inequality.

Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities

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.