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Let X1, . . .,Xn be independent random variables having as common distribution the standard Gaussian measure μ on ℝ2 and let $$ {\mu}_n=\frac{1}{n}\sum \limits_{i=1}^n{\delta}_{X_i} $$ be the associated empirical measure. We show that $$ \frac{1}{C}\frac{\log n}{n}\le $$ 𝔼 $$ \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} $$ for some numerical constant C > 0, where W2 is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan.

We explore upper bounds on Kantorovich transport distances between probability measures on the Euclidean spaces in terms of their Fourier-Stieltjes transforms, with focus on non-Euclidean metrics. The results are illustrated on empirical measures in the optimal matching problem on the real line.

Let $${\widetilde{H}}_N$$ , $$N \ge 1$$ , be the point-to-point last passage times of directed percolation on rectangles $$[(1,1), ([\gamma N], N)]$$ in $${\mathbb {N}}\times {\mathbb {N}}$$ over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some $$\alpha _{\sup } >0$$ , $$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$ with probability one, and that $$\alpha _{\sup } = \big ( \frac{3}{4} \big )^{2/3}$$ provided a commonly believed tail bound holds. The result is in contrast with the normalization $$(\log N)^{2/3}$$ for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result on the liminf with speed $$(\log \log N)^{1/3}$$ is also discussed.

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 (Geom Funct Anal 28:1548–1596, 2018).

We present a new approach, inspired by Stein's method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions. The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erd\H{o}s and Yau., Comment: 72 pages. v2: Minor corrections, a section about the connection with the existing literature added. v3: Change in title

We connect recent developments on Gaussian noise estimation and the Minimum Mean-Square Error to earlier results on entropy and Fisher information heat flow expansion. In particular, the derivatives of the Minimum mean-square error with respect to the noise parameter are related to the heat flow derivatives of the Fisher information and a special Lie algebra structure on iterated gradients. The results lead in particular to a partial answer to the Minimum mean-square error conjecture.

This article is a continuation of the papers [8,9] in which the optimal matching problem, and the related rates of convergence of empirical measures for Gaussian samples are addressed. A further step in both the dimensional and Kantorovich parameters is achieved here, proving that, given $X_1, \ldots, X_n$ independent random variables with common distribution the standard Gaussian measure $\mu$ on $\mathbb{R}^d$, $d \geq 3$, and $\mu_n \, = \, \frac 1n \sum_{i=1}^n \delta_{X_i}$ the associated empirical measure, $$ \mathbb{E} \big( \mathrm {W}_p^p (\mu_n , \mu )\big ) \, \approx \, \frac {1}{n^{p/d}} $$ for any $1\leq p < d$, where $\mathrm {W}_p$ is the $p$-th Kantorovich metric. The proof relies on the pde and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan in a compact setting.

We develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetric-type Harnack inequality is emphasized. Commutation properties between the heat and Hopf-Lax semigroups are developed consequently, providing direct access to the heat flow contraction property along Wasserstein distances.

We address the expected supremum of a linear combination of shifts of the sinc kernel with random coefficients. When the coefficients are Gaussian, the expected supremum is of order \sqrt{\log n}, where n is the number of shifts. When the coefficients are uniformly bounded, the expected supremum is of order \log\log n. This is a noteworthy difference to orthonormal functions on the unit interval, where the expected supremum is of order \sqrt{n\log n} for all reasonable coefficient statistics., To appear in the Journal of Fourier Analysis and Applications

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

International audience; Besides fractional Brownian motion most non-Gaussian fractional fields are obtained by integration of deterministic kernels with respect to a random infinitely divisible measure. In this paper, generalized shot noise series are used to obtain approximations of most of these fractional fields, including linear and harmonizable fractional stable fields. Almost sure and $L^r$-norm rates of convergence, relying on asymptotic developments of the deterministic kernels, are presented as a consequence of an approximation result concerning series of symmetric random variables. When the control measure is infinite, normal approximation has to be used as a complement. The general framework is illustrated by simulations of classical fractional fields.

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., Comment: 22 pages

In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.

The purpose of this study was to evaluate treatment outcomes with intradiscal injection of corticosteroids (IDIC) in cervical spondylotic radiculopathy. Twenty consecutive patients were treated with intradiscal injection of 25 mg of acetate of prednisolone under fluoroscopic control. All patients had previously received a nonsurgical treatment for at least 3 months without success. Outcomes were assessed 1, 3 and 6 months after IDIC. Radicular pain reduction as scored on a visual analogue scale (VAS 100-mm length) was statistically significant at 1 month (19.0+/-28.0 mm; p=0.008), 3 months (25.2+/-27.5 mm; p=0.002), and 6 months (24.6+/-28.4 mm; p=0.001). In all, 40% of treated patients described at least 50% pain improvement 6 months after treatment. Four patients had complete relief of radicular pain. In conclusion, IDIC should be an alternative in the nonsurgical management of cervical spondylotic radiculopathy.

International audience; We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls.

We present a one-dimensional version of the functional form of the geometric Brunn–Minkowski inequality in free (non-commutative) probability theory. The proof relies on matrix approximation as used recently by Biane and Hiai et al. to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered here from the Brunn–Minkowski inequality as in the classical case. The method is used to extend to the free setting the Otto–Villani theorem stating that the logarithmic Sobolev inequality implies the transportation cost inequality. To cite this article: M. Ledoux, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

We study Lipschitz contraction properties of general Markov kernels seen as operators on spaces of probability measures equipped with entropy-like ``distances''. Universal quantitative bounds on the associated ergodic constants are deduced from Dobrushin's ergodic coefficient. Strong contraction properties in Orlicz spaces for relative densities are proved under more restrictive mixing assumptions. We also describe contraction bounds in the entropy sense around arbitrary probability measures by introducing a suitable Dirichlet form and the corresponding modified logarithmic Sobolev constants. The interest in these bounds is illustrated on the example of inhomogeneous Gaussian chains. In particular, the existence of an invariant measure is not required in general.

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 $(E, \mu, \Gamma)$ in which abstract Malliavin-type arguments may be developed and extend the Wiener chaos setting., Comment: 52 pages

This paper is concerned with a general class of self-interacting diffusions {X t } t ≥0 living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX t = Brownian increments + drift term depending on X t and μ t , the normalized occupation measure of the process. It is proved that the asymptotic behavior of {μ t } can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Φ = {Φ t } t ≥0 defined on the space of the Borel probability measures on M. In particular, the limit sets of {μ t } are proved to be almost surely attractor free sets for Φ. These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {μ t } can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction.

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton–Jacobi equations. By the infimum-convolution description of the Hamilton–Jacobi solutions, this approach provides a clear view of the connection between logarithmic Sobolev inequalities and transportation cost inequalities investigated recently by F. Otto and C. Villani. In particular, we recover in this way transportation from Brunn–Minkowski inequalities and for the exponential measure.

We develop several applications of the Brunn—Minkowski inequality in the Prekopa—Leindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Prekopa—Leindler theorem the Brascamp—Lieb inequality for strictly convex potentials. We deduce similarly the logarithmic Sobolev inequality for uniformly convex potentials for which we deal more generally with arbitrary norms and obtain some new results in this context. Applications to transportation cost and to concentration on uniformly convex bodies complete the exposition.

We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting.

This paper describes some of the analytic tools developed recently by Ghirlanda and Guerra in the investigation of the distribution of overlaps in the Sherrington–Kirkpatrick spin glass model and of Parisi's ultrametricity. In particular, we introduce to this task a simplified (but also generalized) model on which the Gaussian analysis is made easier. Moments of the Hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps. Under the essential tool of self-averaging, we describe with full rigour, various overlap identities and replica independence that actually hold in a rather large generality. The results are presented in a language accessible to probabilists and analysts.

— Let M be a complete n-dimensional Riemanian manifold with non-negative Ricci curvature in which one of the Sobolev inequalities (∫ |f |dv )1/p ≤ C(∫ |∇f |qdv)1/q, f ∈ C∞ 0 (M), 1 ≤ q 1. Moreover, for q > 1, the equality in (1) is attained by the functions (λ + |x|q/(q−1))1−(n/q), λ > 0, where |x| is the Euclidean length of the vector x in IR. We are actually interested here in the geometry of those manifolds M for which one of the Sobolev inequalities (1) is satisfied with the best constant C = K(n, q) of IR. The result of this note is the following theorem. Theorem. Let M be a complete n-dimensional Riemannian manifold with nonnegative Ricci curvature. If one of the Sobolev inequalities (1) is satisfied with C = K(n, q), then M is isometric to IR. The particular case q = 1 (p = n/(n − 1)) is of course well-known. In this case indeed, the Sobolev inequality is equivalent to the isoperimetric inequality ( voln(Ω) )(n−1)/n ≤ K(n, 1)voln−1(∂Ω) where ∂Ω is the boundary of a smooth bounded open set Ω in M . If we let V (x0, s) = V (s) be the volume of the geodesic ball B(x0, s) = B(s) with center x0 and radius s in M , we have d ds voln ( B(s) ) = voln−1 ( ∂B(s) ) . Hence, setting Ω = B(s) in the isoperimetric inequality, we get V (s)(n−1)/n ≤ K(n, 1)V ′(s) for all s. Integrating yields V (s) ≥ (nK(n, 1))−nsn, and since K(n, 1) = n−1ω n , for every s, (2) V (s) ≥ V0(s) where V0(s) = ωns is the volume of the Euclidean ball of radius s in IR. IfM has nonnegative Ricci curvature, by Bishop’s comparison theorem (cf. e.g. [Ch]) V (s) ≤ V0(s) for every s, and by (2) and the case of equality,M is isometric to IR. The main interest of the Theorem therefore lies in the case q > 1. As usual, the classical value q = 2 (and p = 2n/(n − 2)) is of particular interest (see below). It should be noticed that known results already imply that the scalar curvature of M is zero in this case (cf. [He], Prop. 4.10). Proof of the Theorem. It is inspired by the technique developed in the recent work [B-L] where a sharp bound on the diameter of a compact Riemannian manifold satisfying a Sobolev inequality is obtained, extending the classical Myers theorem. We thus assume that the Sobolev inequality (1) is satisfied with C = K(n, q) for some q > 1. Recall first that the extremal functions of this inequality in IR are the functions (λ + |x|q)1−(n/q), λ > 0, where q′ = q/(q − 1). Let now x0 be a fixed point in M and let θ > 1. Set f = θ−1d(·, x0) where d is the distance function on M . The idea is then to apply the Sobolev inequality (1), with C = K(n, q), to (λ+ f ′ )1−(n/q), for every λ > 0 to deduce a differential inequality whose solutions may be compared to the extremal Euclidean case. Set, for every λ > 0, F (λ) = 1 n− 1 ∫ 1 (λ+ fq)n−1 dv. Note first that F is well defined and continuously differentiable in λ. Indeed, by Fubini’s theorem, for every λ > 0

This chapter is a brief investigation of the links between optimal transportation methods and functional inequalities in the Markov operator framework of this monograph. After a brief introduction to the basic material on optimal transportation, the main topic of transportation cost inequalities and first examples for Gaussian measures are presented. Interpolation along the geodesics of optimal transport is used towards logarithmic Sobolev inequalities and transportation cost inequalities comparing relative entropy and Wasserstein distances between probability measures. An alternate approach to sharp Sobolev or Gagliardo–Nirenberg inequalities in Euclidean space is provided next along these lines. Non-linear Hamilton–Jacobi equations and hypercontractivity properties of their solutions, analogous to the ones for linear heat equations, are investigated in the further sections towards the relationships between (quadratic) transportation cost inequalities and logarithmic Sobolev inequalities. Contraction properties in Wasserstein space along with the heat semigroup are investigated in the Markov operator setting. The last section is a very brief overview of recent developments towards a notion of Ricci curvature lower bounds based on optimal transportation and the connection with the Γ-calculus developed in this work.