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We study a $1$-dimensional chain of $N$ weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with $N$ dependent bounds) perturbations of the harmonic ones. We show how a generalized version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by F. Baudoin (2017). By that we prove exponential convergence to non-equilibrium steady state (NESS) in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than $N^{-3}$. For the purely harmonic chain the order of the rate is in $ [N^{-3},N^{-1}]$. This shows that, in this set up, the spectral gap decays at most polynomially with $N$., Comment: 54 pages, 1 figure, comments are welcome

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^*$-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates., An error in Theorem 10.6 has been corrected

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 sequence of Poincaré constants corresponding to successive convolutions in the central limit theorem is non-increasing. 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 (Teor. Veroyatn. Primen. 48 (2003) 615–620). As a byproduct of our arguments, we find that the various monotone properties of entropy, Fisher information and the Poincaré constant on convolutions have a common, simple root in Shearer’s inequality.

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad's angular cut-off assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cut-off case and conjecture what we believe to be the right rate of convergence in that case.

A functional inequality, called supermigrativity, was recently introduced for bivariate semi-copulas and applied in various problems arising in the study of aging properties of stochastic systems. Here, we revisit this notion and extend it to the case of aggregation functions in higher dimensions. In particular, we show how supermigrativity can be expressed via monotonicity of a function with respect to logarithmic majorization ordering of real vectors. Various alternative characterizations of supermigrativity are illustrated, together with some of its weaker versions. Several examples show similarities and differences between the bivariate and the general case.

In this paper we prove the functional inequality $f(x)^{f(x)}\leq g(x)^{g(x)}$ for positive real functions $f$ and $g$ satisfying natural conditions and apply it to derive inequalities between some of the elementary functions and to prove monotonocity of certain sequences of real numbers.

International audience; We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.

The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull $\Omega^*$ of a set $\Omega$ is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where $\Omega$ has $\mathscr{C}^{1, \alpha}$-boundary, the area of $\partial \Omega^*$ is recovered as the limit of the $p$-capacities of $\Omega$, as $p \to 1^+$. Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided $3 \leq n \leq 7$.

This work is related to inequalities in the approximation theory. Mainly, we study numerical solutions of delay Volterra integral equations by using a collocation method based on sigmoidal function approximation. Error estimation and convergence analysis are provided. At the end of the paper we display numerical simulations verifying our results.

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and \'Emery, the exponential curvature-dimension condition of Bauer \textit{et al.} and the coarse Ricci curvature of Ollivier. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_1$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Meyer's theorem under the curvature-dimension condition CD$(\kappa,\infty)$ of Bakry and \'Emery., Comment: 26 pages. Comments are welcome

In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter's sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Tur\'an type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter's paper [Ka] and at the end of the paper we pose an open problem, which may be of interest for further research., Comment: 8 pages

We prove the following theorem. Let α and β be real numbers. The inequality Γ ( x α + y β ) ≤ Γ ( Γ ( x ) + Γ ( y ) ) holds for all positive real numbers x and y if and only if α = β = − γ . Here, Γ and γ = 0.57721 … denote Euler’s gamma function and Euler’s constant, respectively.

We present several inequalities for f a ( x ) = Γ ( a , x ) Γ ( a , 0 ) ( a > 0 , x ⩾ 0 ) , where Γ ( a , x ) is the incomplete gamma function. One of our theorems states that the inequalities f a ( S p ( x 1 , … , x n ) ) ⩽ f a ( x 1 ) ⋯ f a ( x n ) ⩽ f a ( S q ( x 1 , … , x n ) ) ( p , q > 0 ) hold for all nonnegative real numbers x 1 , … , x n ( n ⩾ 2 ) if and only if p ⩽ m i n ( a , 1 ) and q ⩾ m a x ( a , 1 ) . Here, S t ( x 1 , … , x n ) denotes the power sum of order t. This extends and complements a result published by Ismail and Laforgia in 2006.

In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\'an type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further research., Comment: 14 pages

In this paper we extend some results in [Dinh, Goberna, López, and Volle, Set-Valued Var. Anal., to appear] to the setting of functional inequalities when the standard assumptions of convexity and lower semicontinuity of the involved mappings are absent. This extension is achieved under certain condition relative to the second conjugate of the involved functions. The main result of this paper, Theorem 1, is applied to derive some subdifferential calculus rules and different generalizations of the Farkas lemma for nonconvex systems, as well as some optimality conditions and duality theory for infinite nonconvex optimization problems. Several examples are given to illustrate the significance of the main results and also to point out the potential of their applications to get various extensions of Farkas-type results and to the study of other classes of problems such as variational inequalities and equilibrium models. This research was partially supported by MICINN of Spain, grant MTM2008-06695-C03-01.

This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincar\'e inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on $\R_{+}$ with the reference example of the Marcenko-Pastur distribution., Comment: This paper will apear in Journal of Fucntional Analysis

Let (Ω, Σ, μ) be a measure space and ϕ, ψ : (0, ∞) → (0, ∞) some bijective functions. Suppose that the functional Pϕ,ψ defined on class of μ-integrable simple functions χ : Ω → [0, ∞), μ({ϖ : χ(ϕ) > 0} > 0, by the formula Pϕ,ψ(χ) = ψ∫{χ>0}ϕo χdμ satisfies the triangle inequality. We prove that if there are A, B ϵ Σ such that 0 < μ(A) < 1 < μ(B) < ∞, the function ψ o ϕ is superadditive, and limt → 0 ψ(t) = 0 then there is a p ≥ 1 such that ϕ(t)=ϕ(1)tp, ψ(t)=ψ(1)t1p, t>0), The assumption limt → 0 ψ(t) = 0 can be significantly weakened or, for some measure spaces, even omitted. The remaining assumptions are essential. In particular, in each of the cases: (i) A ϵ Σ ⇒ μ(A) = 0 or μ(A ≥ 1; (ii) A ϵ Σ ⇒ μ(A) ≤ 1 or μ(A) = ∞, some broad classes of pairs (ϕ, ψ) of non-power functions for which Pϕ,ψ is subadditive are indicated. These results give a solution of an open problem posed by W. Wnuk. The reversed triangle inequality is also considered.

Agradecimentos: This work was partially done while visiting the Isaac Newton Institute, Cambridge, UK, and the authors are grateful for its hospitality. J.A.C. acknowledges support from projects MTM2011-27739-C04-02, 2009-SGR-345 from Agencia de Gestio d'Ajuts Universitaris i de Recerca-Generalitat de Catalunya, and the Royal Society through a Wolfson Research Merit Award. J.A.C. and Y.H. acknowledge support from the Engineering and Physical Sciences Research Council (UK) grant number EP/K008404/1. M.C.S. acknowledges support from Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior - CAPES (BR) project number BEX 0024/13-9. J.L.V. is partially supported by Spanish Project MTM2011-24696 Abstract: We analyse the asymptotic behaviour of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and Vazquez [1,2], where the pressure is obtained as a Riesz potential associated with the density. We take advantage of the displacement convexity of the Riesz potential in one dimension to show a functional inequality involving the entropy, entropy dissipation, and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence of solutions in self-similar variables to the unique steady states COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPES Fechado

International audience; We study optimal functions in a family of Caffarelli-Kohn-Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.

We prove an optimal diffusive decay of the environment viewed by the particle in random walk among random independent conductances, with, as a main assumption, finite second moment of the conductance. Our proof, using the analytic approach of Gloria, Neukamm and Otto, is very short and elementary., Comment: 13 pages

International audience; We describe conditions on non-gradient drift diffusion Fokker-Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, which to our knowledge is a novelty.

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell's formula for the Laplace transform. As an application, we give unified and short proofs of a number of functional inequalities., Comment: 18 pages

Given a birth-death process on $\mathbb {N}$ with semigroup $(P_t)_{t\geq0}$ and a discrete gradient ${\partial}_u$ depending on a positive weight $u$, we establish intertwining relations of the form ${\partial}_uP_t=Q_t\,{\partial}_u$, where $(Q_t)_{t\geq0}$ is the Feynman-Kac semigroup with potential $V_u$ of another birth-death process. We provide applications when $V_u$ is nonnegative and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ433 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We give a direct proof of a functional Santalo inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santalo inequality. The argument combines a logarithmic form of the Prekopa-Leindler inequality and a partition theorem of Yao and Yao., Comment: 6 pages, file might be slightly different from the published version

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\'e constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincar\'e inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions., Comment: Corrections. To appear in Annales de l'Institut Henri Poincare (AIHP)

Accepted for publication in ESAIM/PS.; International audience; This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\infty$ queue. They describe in particular the exponential dissipation of $\Phi$-entropies along this process. This simple queueing process appears as a model of ``constant curvature'', and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/$\infty$ queues. Proofs are elementary and rely essentially on the development of a ``$\Phi$-calculus''.

We show that a strong mixing condition implies a Bakry-Bobkov-Ledoux inequality for a probability measure on infinite-dimensional space.

In this paper, by defining a fuzzy inner product on the dual of real fuzzy Hilbert space, we show that the dual of fuzzy Hilbert space is also fuzzy Hilbert space. Then the concepts of fuzzy linear operator and fuzzy isomentric are defined. It is shown that each real fuzzy Hilbert space is isometricaly isomorphic to the dual of the fuzzy Hilbert space. Publisher's Version

In this work, we apply several functional inequalities (Poincaré,logarithmic Sobolev etc.) to solve two problems. First, we study aninhomogeneous diffusion, akin to the discrete simulated annealingalgorithm, in the spirit of a paper by L. Miclo. We show that thisdiffusion converges under weaker hypotheses than what was assumed inprevious work. In particular, the potential governing the drift isallowed to grow very slowly at infinity. In a second part, we turn toa model of statistical mechanics with unbounded spins, recentlystudied by T. Bodineau and B. Helffer, N. Yoshida and G. Royer (amongothers). We clarify links between mixing properties, uniqueness of theGibbs measure, and functional inequalities. We show in particular thatthe infinite volume tempered Gibbs measure is unique, provided thatthe finite volume measures for one boundary condition satisfy aBeckner inequality.; Dans cette thèse, nous utilisons différentes inégalités fonctionnelles(Poincaré, Sobolev logarithmique, etc.) pour étudier deux questions.Nous appliquons d'abord des inégalités affaiblies à l'étude d'unediffusion inhomogène, analogue continu de l'algorithme de recuitsimulé, dans la lignée d'un travail de L. Miclo. Nous montrons unrésultat de convergence de la diffusion, sous des hypothèses plusfaibles que celles posées précédemment : le potentiel dans lequel ladiffusion évolue peut croître très lentement à l'infini.Dans le cadre d'un modèle de mécanique statistique à spins non-bornés,en nous basant sur des résultats de T. Bodineau et B. Helffer, N. Yoshidaet G. Royer, nous éclaircissons ensuite les liens entre différentesinégalités fonctionnelles, des propriétés de mélange et l'unicité dela mesure de Gibbs en volume infini. Nous montrons en particulierl'unicité si les mesures en volume fini et pour une seule conditionaux bords vérifient uniformément une inégalité de Beckner.

In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some ''curvature'' assumptions. Our notion of curvature coincides with the usual $\Gamma_2$ curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.