arXiv subjects

Ivan Gentil

Publications and source records attributed to Ivan Gentil.

At least 19 recordsLinked to original sources

arXiv2022

Time reversal of diffusion processes under a finite entropy condition

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 wide class of diffusion processes in $ \mathbb{R}^n$ possibly with singular drifts, extending the already known results in this domain. The proof of the integration by parts formula relies on stochastic derivatives. Then, this formula is applied to compute the semimartingale characteristics of the time-reversed $P^*$ of a diffusion measure $P$ provided that the relative entropy of $P$ with respect to another diffusion measure $R$ is finite, and the semimartingale characteristics of the time-reversed $R^*$ are known (for instance when the reference path measure $R$ is reversible). As an illustration of the robustness of this method, the integration by parts formula is also employed to derive a time-reversal formula for a random walk on a graph.

math.PR↗

arXiv2022

Stability estimates for the sharp spectral gap bound under a curvature-dimension condition

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$(N-1, N)$ space is almost minimal, then the pushforward of the measure by an eigenfunction associated with the spectral gap is close to a Beta distribution. The proof combines estimates on the eigenfunction obtained via a new $L^1$-functional inequality for RCD spaces with Stein's method for distribution approximation. We also derive analogous, almost sharp, estimates for infinite and negative values of the dimension parameter.

math.MG↗

arXiv2021

A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality

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, we prove that the CKN inequality can be interpreted in this way on three different and equivalent models, obtained as weighted versions of the standard Euclidean space, round sphere and hyperbolic space. This result can be viewed as an extension of conformal invariance to the weighted setting. Since the spherical CKN model we introduce has finite measure, the $\Gamma$-calculus introduced by Bakry and Emery provides an easy way to prove the Sobolev inequalities. This method allows us to recover the optimality of the region of parameters describing symmetry-breaking of minimizers of the CKN inequality, introduced by Felli and Schneider and proved by Dolbeault, Esteban and Loss in 2016. Finally, we develop the notion of n-conformal invariants, exhibiting a way to extend the notion of scalar curvature to weighted manifolds such as the CKN models.

math.AP↗

arXiv2021

On the variational interpretation of local logarithmic Sobolev inequalities

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 Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $Γ$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schr{ö}dinger problem yields a variations interpretation of the local logarithmic Sobolev inequalities, that could possibly unlock novel class of local inequalities.

math.CA↗

arXiv2021

Long-time behaviour of entropic interpolations

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 Brownian particles conditionally to observations. It is well known that in the short time limit entropic interpolations converge to the McCann-geodesics of optimal transport. Here we focus on the long-time behaviour, proving in particular asymptotic results for the entropic cost and establishing the convergence of entropic interpolations towards the heat equation, which is the gradient flow of the entropy according to the Otto calculus interpretation. Explicit rates are also given assuming the Bakry-{É}mery curvature-dimension condition. In this respect, one of the main novelties of our work is that we are able to control the long time behavior of entropic interpolations assuming the CD(0, n) condition only.

math.AP↗

arXiv2021

Sobolev's inequality under a curvature-dimension condition

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 inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut-V{é}ron and V{é}ron's 1991 classical work.

math.AP↗

arXiv2020

The entropy, from Clausius to functional inequalities

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 concept of entropy. Instead of simply defining a functional, central point for the development of the second principle, it will in fact define a concept sufficiently general to be used in many fields of mathematics.In these few pages, I wish to show the role played by entropy in the field of gradient flows and functional inequalities. Starting from the definition of Clausius in 1865, I will try to explain how fundamental functional inequalities like Sobolev inequality, key point in analysis, are natural inked with a particular entropy. This path allows me to give an overview of the use of gradient flows in finite or infinite dimension, of Bakry-Emery theory and more recently of Otto's calculus.

math.PR↗

arXiv2019

The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities

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 rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent Brownian particles. We undertake a fine study of the dynamics of its solutions, including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the the time between observations grows larger and larger, as well as a novel class of functional inequalities involving the mean field entropic cost (i.e. the optimal value in (MFSP)). Our strategy unveils an interesting connection between forward backward stochastic differential equations and the Riemannian calculus on the space of probability measures introduced by Otto, which is of independent interest.

math.PR↗

arXiv2019

A family of Beckner inequalities under various curvature-dimension conditions

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) condition, either with $ρ$ = 0 and n > 0, or with $ρ$ > 0 and n $\in$ R. As such, results are valid in the case of a Riemannian manifold, which constitutes a generalization to what was proved in [BGS18, Ngu18].

math.FA↗

arXiv2018

Sharp Beckner-type inequalities for Cauchy and spherical distributions

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 Cauchy type distributions on the Euclidean space. These optimal inequalities appear to be equivalent to some non tight optimal Beckner inequalities on the sphere, and the family appear to be a new form of the Sobolev inequality.

math.PR↗

arXiv2018

An entropic interpolation proof of the HWI inequality

We present a pathwise proof of the HWI inequality which is based on en-tropic interpolations rather than displacement ones. Unlike the latter, entropic interpolations are regular both in space and time. Consequently, our approach is closer to the Otto-Villani heuristics, presented in the first part of the article [23], than the original rigorous proof presented in the second part of [23].

math.PR↗

arXiv2018

Dynamical aspects of generalized Schr{ö}dinger problem via Otto calculus -- A heuristic point of view

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 slowed down gradient flow equations: $\dot ω^{ \varepsilon}\_t=- \varepsilon F'( ω^{ \varepsilon}\_t),$ where $\varepsilon>0$, and (ii) by considering the \emph{accelerations} $\ddot ω^{ \varepsilon}\_t$. We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus.A special formulation of the Schr{ö}dinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying Schr{ö}dinger problem, with a general function of the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when some fluctuation parameter tends to zero. We show heuristically that the solutions satisfy a Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequality under curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schr{ö}dinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.

math.PR↗

arXiv2017

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.

math.PR↗

arXiv2017

New sharp Gagliardo-Nirenberg-Sobolev inequalities and an improved Borell-Brascamp-Lieb inequality

We propose a new Borell-Brascamp-Lieb inequality which leads to novel sharp Euclidean inequalities such as Gagliardo-Nirenberg-Sobolev inequalities in R^n and in the half-space R^n\_+. This gives a new bridge between the geometric pont of view of the Brunn-Minkowski inequality and the functional point of view of the Sobolev type inequalities. In this way we unify, simplify and results by S. Bobkov-M. Ledoux, M. del Pino-J. Dolbeault and B. Nazaret.

math.PR↗

arXiv2016

The Li-Yau inequality and applications under a curvature-dimension condition

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

math.DG↗

arXiv2016

About the analogy between optimal transport and minimal entropy

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 representation formula of the entropic cost are derived, as well as contraction inequalities with respect to the entropic cost. This analogy is also illustrated with some numerical examples where the reference path measure is given by the Brownian or the Ornstein-Uhlenbeck process. Our point of view is measure theoretical and the relative entropy with respect to path measures plays a prominent role.

math.PR↗

arXiv2015

Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition

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 semigroup, and quadratic Wasserstein distance contraction properties at different times. On the other hand, in a compact Riemannian manifold, it implies a same-time Wasserstein contraction property for this semigroup. In this work we generalize the latter result to metric measure spaces and more importantly prove the converse: contraction inequalities are equivalent to curvature-dimension conditions. Links with functional inequalities are also investigated.

math.PR↗

arXiv2014

Dimensional contraction in Wasserstein distance for diffusion semigroups on a Riemannian manifold

We prove a refined contraction inequality for diffusion semigroups with respect to the Wasserstein distance on a compact Riemannian manifold taking account of the dimension. The result generalizes in a Riemannian context, the dimensional contraction established in [BGG13] for the Euclidean heat equation. It is proved by using a dimensional coercive estimate for the Hodge-de Rham semigroup on 1-forms.

math.PR↗