Ivan Gentil
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View article: Stability estimates for the sharp spectral gap bound under a curvature-dimension condition
Stability estimates for the sharp spectral gap bound under a curvature-dimension condition Open
We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing that if the spectral gap of an space is a…
Sobolev’s inequality under a curvature-dimension condition Open
In this note we present a new proof of Sobolev’s inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous …
View article: Stability estimates for the sharp spectral gap bound under a curvature-dimension condition
Stability estimates for the sharp spectral gap bound under a curvature-dimension condition Open
We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing that if the spectral gap of an RCD$(N-1, …
A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality Open
We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short), introduced by these authors in 1984. We explain why the CKN inequality can be viewed as a Sobolev inequality on a weighted Riemannian manifold. More precisely, w…
Time reversal of diffusion processes under a finite entropy condition Open
Motivated by entropic optimal transport, time reversal of diffusion processes is revisited. An integration by parts formula is derived for the carré du champ of a Markov process in an abstract space. It leads to a time reversal formula for…
A family of Beckner inequalities under various curvature-dimension conditions Open
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Sobolev's inequality under a curvature-dimension condition Open
In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous …
On the variational interpretation of local logarithmic Sobolev inequalities Open
The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic S…
The entropy, from Clausius to functional inequalities Open
In this document we are interested in entropy. Entropy is multiple, the idea is to describe the definition proposed by the physicist Clausius. Indeed, Clausius exposes in 1865 the second principle of thermodynamics and also proposes the co…
Long-time behaviour of entropic interpolations Open
In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schr{\"o}dinger problem [Sch31], which is the problem of finding the most likely evolution of a system of independent…
Long-time behaviour of entropic interpolations Open
In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schr{ö}dinger problem [Sch31], which is the problem of finding the most likely evolution of a system of independent B…
The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities Open
We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian particles conditionally on the observation of their initial and final configuration. Its rigoro…
Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view Open
The defining equation (\ast)\qquad \dot \omega_t=-F'(\omega_t) of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation (\ast) into the family…
Sharp Beckner-type inequalities for Cauchy and spherical distributions Open
Using some harmonic extensions on the upper-half plane, and probabilistic representations, and curvature-dimension inequalities with some negative dimensions, we obtain some new opimal functional inequalities of the Beckner type for the Ca…
The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities Open
We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian particles conditionally on the observation of their initial and final configuration. Its rigoro…
A family of Beckner inequalities under various curvature-dimension\n conditions Open
In this paper, we offer a proof for a family of functional inequalities\ninterpolating between the Poincar{\\'e} and the logarithmic Sobolev (standard\nand weighted) inequalities. The proofs rely both on entropy flows and on a\nCD($\\rho$,…
A family of Beckner inequalities under various curvature-dimension conditions Open
In this paper, we offer a proof for a family of functional inequalities interpolating between the Poincar{é} and the logarithmic Sobolev (standard and weighted) inequalities. The proofs rely both on entropy flows and on a CD($ρ$, n) condit…
AN ENTROPIC INTERPOLATION PROOF OF THE HWI INEQUALITY Open
The HWI inequality is an "interpolation"inequality between the Entropy H, the Fisher information I and the Wasserstein distance W. We present a pathwise proof of the HWI inequality which is obtained through a zero noise limit of the Schröd…
Dynamical aspects of generalized Schr{ö}dinger problem via Otto calculus -- A heuristic point of view Open
The defining equation $(\ast):\ \dot ω\_t=-F'(ω\_t),$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the family of slo…
Dynamical aspects of generalized Schr{\\"o}dinger problem via Otto\n calculus -- A heuristic point of view Open
The defining equation $(\\ast):\\ \\dot \\omega\\_t=-F'(\\omega\\_t),$ of a gradient\nflow is kinetic in essence. This article explores some dynamical (rather than\nkinetic) features of gradient flows (i) by embedding equation $(\\ast)$ in…
Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition Open
The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigro…
New Sharp Gagliardo–Nirenberg–Sobolev Inequalities and an Improved Borell–Brascamp–Lieb Inequality Open
We propose a new Borell–Brascamp–Lieb inequality that leads to novel sharp Euclidean inequalities such as Gagliardo–Nirenberg–Sobolev inequalities in $ {\mathbb{R}}^n$ and in the half-space $ {\mathbb{R}}^n_+$. This gives a new bridge betw…
Sharp Beckner-type inequalities for Cauchy and spherical distributions Open
Using some harmonic extensions on the upper-half plane, probabilistic representations, and curvature-dimension inequalities with negative dimensions, we obtain some new optimal functional inequalities of Beckner type for Cauchy-type distri…
When Otto meets Newton and Schrödinger, an heuristic point of view Open
We propose a generalization of the Schr\"odinger problem by replacing the usual entropy with a functional $\mathcal F$ which approaches the Wasserstein distance along the gradient of $\mathcal F$. From an heuristic point of view by using O…
Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities Open
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 a…
Solution of a class of reaction-diffusion systems via logarithmic Sobolev inequality Open
We study global existence, uniqueness and positivity of weak solutions of a class of reaction-diffusion systems coming from chemical reactions. The principal result is based only on a logarithmic Sobolev inequality and the exponential inte…
About the analogy between optimal transport and minimal entropy Open
We describe some analogy between optimal transport and the Schrödinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type formulation and a Benamou–Brenier type represent…