Logarithmic Sobolev Inequality Research Articles

Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions

We prove a version of Talagrand’s concentration inequality for subordinated sub-Laplacians on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite-dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.

Classification of solutions of an equation related to a conformal log Sobolev inequality

We classify all finite energy solutions of an equation which arises as the Euler–Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving spheres from Rn to Sn and a classification result of Li and Zhu. Along the way we prove a small volume maximum principle and a strong maximum principle for the underlying operator which is closely related to the logarithmic Laplacian.

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

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 more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two.

Global Existence and Blow-Up for the Fractional p-Laplacian with Logarithmic Nonlinearity

In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity $$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+(-\Delta )^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} {\mathbb {R}}^{N}\backslash \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N \, ( N\ge 1)$$ is a bounded domain with Lipschitz boundary and $$2\le p< \infty $$ . The local existence will be done using the Galerkin approximations. By combining the potential well theory with the Nehari manifold, we establish the existence of global solutions. Then by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.

Multiscale functional inequalities in probability: Constructive approach

Consider an ergodic stationary random field A on the ambient space ℝ d . In order to establish concentration properties for nonlinear functions Z(A), it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.

On the diffusion p(x)-Laplacian with logarithmic nonlinearity

In this paper, we consider the following class of heat equation involving p(x)-Laplacian with logarithmic nonlinearity $$\begin{aligned}\left\{ \begin{array}{llc} u_{t}-\Delta _{p(x)}u=|u|^{s(x)-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} \partial \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^{N}$$ ( $$N\ge 1$$ ) is a bounded domain with smooth boundary $$\partial \Omega $$ , $$p, s: \overline{\Omega }\rightarrow \mathbb {R}_{+}$$ are continuous functions that satisfy some technical conditions and $$-\Delta _{p(x)}$$ is the $$p(x)-$$ Laplacian, which generalizes the $$p-$$ Laplacian operator $$-\Delta _{p}$$ . The local existence will be done by using the Galerkin method. Then, by using the concavity method we prove that the local solutions blow-up in finite time under suitable conditions. In order to prove the global existence, we will use the potential well theory combined with the Pohozaev manifold that is a novelty for this type of problem. The difficulty here is the lack of logarithmic Sobolev inequality which seems there is no logarithmic Sobolev inequality concerning the p(x)-Laplacian yet.

Second- and Third-Order Asymptotics of the Continuous-Time Poisson Channel

The paper derives the optimal second-order coding rate for the continuous-time Poisson channel. We also obtain bounds on the third-order coding rate. This is the first instance of a second-order result for a continuous-time channel. The converse proof hinges on a novel construction of an output distribution induced by Wyner's discretized channel and the construction of an appropriate $\epsilon$-net of the input probability simplex. While the achievability proof follows the general program to prove the third-order term for non-singular discrete memoryless channels put forth by Polyanskiy, several non-standard techniques -- such as new definitions and bounds on the probabilities of typical sets using logarithmic Sobolev inequalities -- are employed to handle the continuous nature of the channel.

Rigidity of some functional inequalities on RCD spaces

We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Émery inequality. As an application, we prove the rigidity and identify the extremal functions of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincaré inequality in the setting of RCD(K,∞) metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce [19], Morgan [44], Bouyrie [18], Ohta-Takatsu [45], Cheng-Zhou [23].Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandrov spaces with curvature lower bound. Some results including the rigidity of the 1-Bakry-Émery inequality, the rigidity of Φ-entropy inequalities are of particular interest even in the smooth setting.

The Equivalence of Hypercontractivity and Logarithmic Sobolev Inequality for q (−1 ≤ q ≤ 1) -Ornstein-Uhlenbeck Semigroup

In this paper the author proves the equivalence of hypercontractivity and logarithmic Sobolev inequality for q-Ornstein-Uhlenbeck semigroup U() = Γq(e−tI) (−1 ≤ q ≤ 1), where Γq is a q-Gaussian functor.

Stability of the logarithmic Sobolev inequality via the Föllmer process

On étudie les propriétés de stabilité et d’instabilité de l’inégalité de Sobolev logarithmique gaussienne, en termes de covariance, de distance de Wasserstein et d’information de Fisher, répondant à plusieurs questions ouvertes dans la littérature. On établit d’abord une forme améliorée de l’inégalité de Sobolev logarithmique qui est à la fois invariante par transformation linéaire et indépendante de la dimension. Comme corollaire, on obtient une inégalité de stabilité optimale et indépendante de la dimension pour les mesures dont la covariance est majorée par l’identité. On se penche ensuite sur la question de savoir dans quelle mesure le déficit dans l’inégalité de Sobolev logarithmique contrôle la covariance de la mesure. On montre notamment que si la covariance est majorée par l’identité, alors à covariance fixée, la mesure gaussienne minimise ce déficit. D’un autre côté on présente un contrexemple montrant que sans hypothèse de covariance bornée, l’inégalité est instable. Enfin, on étudie la question de la stabilité en termes de distance de Wasserstein, et on montre que même en se restreignant aux mesures dont la covariance est bornée, il n’est pas possible d’obtenir un résultat de stabilité qui soit indépendant de la dimension. Les contrexemples que nous exhibons suggèrent une nouvelle notion de stabilité, en terme de proximité de la mesure à un mélange de gaussiennes. On démontre plusieurs résultats dans cette direction, certains étant indépendants de la dimension. Ces résultats sont par ailleurs plus forts que certains résultats de stabilité qu’on trouve dans la littérature. Nos techniques de preuve reposent fortement sur des méthodes stochastiques.

Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity

In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global well-posedness of strong solutions is established for any initial data such that the initial horizontal velocity v0∈H2(Ω) and the initial temperature T0∈H1(Ω)∩L∞(Ω) with ∇HT0∈Lq(Ω), for some q∈(2,∞). Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to H2(Ω). The main difficulties are the absence of the vertical viscosity and the lack of the horizontal diffusivity, which, interact with each other, thus causing the “mismatching” of regularities between the horizontal momentum and temperature equations. To handle this “mismatching” of regularities, we introduce several auxiliary functions, i.e., η,θ,φ, and ψ in the paper, which are the horizontal curls or some appropriate combinations of the temperature with the horizontal divergences of the horizontal velocity v or its vertical derivative ∂zv. To overcome the difficulties caused by the absence of the horizontal diffusivity, which leads to the requirement of some Lt1(Wx1,∞)-type a priori estimates on v, we decompose the velocity into the “temperature-independent” and temperature-dependent parts and deal with them in different ways, by using the logarithmic Sobolev inequalities of the Brézis–Gallouet–Wainger and Beale–Kato–Majda types, respectively. Specifically, a logarithmic Sobolev inequality of the limiting type, introduced in our previous work (Cao et al., 2016), is used, and a new logarithmic type Gronwall inequality is exploited.

Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

In this paper, we prove multilevel concentration inequalities for bounded functionals f = f(X_1, ldots , X_n) of random variables X_1, ldots , X_n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes f(X) = sup _{g in {mathcal {F}}} {|g(X)|} and suprema of homogeneous chaos in bounded random variables in the Banach space case f(X) = sup _{t} {Vert sum _{i_1 ne ldots ne i_d} t_{i_1 ldots i_d} X_{i_1} cdots X_{i_d}Vert }_{{mathcal {B}}}. The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U-statistics with bounded kernels h and for the number of triangles in an exponential random graph model.

Logarithmic Sobolev inequalities for finite spin systems and applications

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but around a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdős–Rényi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution

We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

Noncollapsing estimate for the Ricci-Bourguignon flow

The Ricci-Bourguignon flow (R-B flow) is a general geometric evolving equation, which includes or relates to some famous geometric flows, for example the Ricci flow and the Yamabe flow, etc. In this paper we shall prove that for the R-B flow evolving on [0,T), whose first eigenvalue λ0 of the operator −△+(1−(n−1)ρ)24(1−2(n−1)ρ)R for the initial metric g(0) is positive, or T>0 is finite, an upper bound assumption of the scalar curvature implies a noncollapsing estimate of the volume, uniformly for all time. In order to derive this noncollapsing estimate, we firstly establish a logarithmic Sobolev inequality along the R-B flow, by using the monotone formula for the Perelman's functional, and then we can derive a Sobolev inequality along the R-B flow.

Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality

Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation cost inequalities have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure and convexity theory. Here, we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one.

Around the entropic Talagrand inequality

In this article, we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of inequalities has been introduced in the recent work (Probab. Theory Related Fields 174 (2019) 1–47), in connection with the study of Schrödinger bridges. We provide several equivalent characterizations in terms of reverse hypercontractivity for the heat semigroup, contractivity of the Hamilton–Jacobi–Bellman semigroup and dimension-free concentration of measure. Properties such as tensorization and relations to other functional inequalities are also investigated. In particular, we show that the inequalities studied in this article are implied by a Logarithmic Sobolev inequality and imply Talagrand inequality.

Global well-posedness for the nonlinear damped wave equation with logarithmic type nonlinearity

The initial boundary value problem for the nonlinear wave equations with damping and logarithmic nonlinearity is investigated in this paper. By making use of modified potential well theory and the technique of Logarithmic-Sobolev inequality, we establish global existence as well as asymptotic behavior of solution, under the assumption that the initial energy is small. Moreover, we obtain an exponential decay which is much faster than the decay in polynomial nonlinear case of Gazzola and Squassina (Ann I H Poincare AN 23:185–207, 2006). These results generalize and extend work in application of potential well theory to wave equations.

Uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions

ABSTRACT The aim of this article is to formulate some new uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of Pitt's inequality for the continuous shearlet transforms, then we formulate Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner inequality in the Fourier domain. In continuation, a version of the logarithmic Sobolev inequality having a dual relation with Beckner's inequality is obtained. In sequel, the Nazarov's uncertainty principle is also derived for the continuous shearlet transforms in arbitrary space dimensions. The article concludes with the formulation of certain new local type uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions.

Uniform LSI for the canonical ensemble on the 1D-lattice with strong, finite-range interaction

We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show that the canonical ensemble (ce) satisfies a uniform logarithmic Sobolev inequality (LSI). The LSI constant is uniform in the boundary data, the external field and scales optimally in the system size. This extends a classical result of H.T. Yau from discrete to unbounded, real-valued spins. It also extends prior results of Landim et al. or Menz for unbounded, real-valued spins from absent- or weak- to strong-interaction. We deduce the LSI by combining two competing methods, the two-scale approach and the Zegarlinski method. Main ingredients are the strict convexity of the coarse-grained Hamiltonian, the equivalence of ensembles and the decay of correlations in the ce.