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dc.contributor.authorEldan, Ronen
dc.contributor.authorLehec, Joseph
dc.contributor.authorShenfeld, Yair
dc.date.accessioned2019-12-18T10:51:01Z
dc.date.available2019-12-18T10:51:01Z
dc.date.issued2019
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/20335
dc.language.isoenen
dc.subjectSobolev inequalityen
dc.subjectFöllmer Processen
dc.subject.ddc519en
dc.titleStability of the logarithmic Sobolev inequality via the Föllmer Processen
dc.typeDocument de travail / Working paper
dc.description.abstractenWe study the stability and instability of the Gaussian logarithmic Sobolev inequality, in terms of covariance, Wasserstein distance and Fisher information, addressing several open questions in the literature. We first establish an improved logarithmic Sobolev inequality which is at the same time scale invariant and dimension free. As a corollary, we show that if the covariance of the measure is bounded by the identity, one may obtain a sharp and dimension-free stability bound in terms of the Fisher information matrix. We then investigate under what conditions stability estimates control the covariance, and when such control is impossible. For the class of measures whose covariance matrix is dominated by the identity, we obtain optimal dimension-free stability bounds which show that the deficit in the logarithmic Sobolev inequality is minimized by Gaussian measures, under a fixed covariance constraint. On the other hand, we construct examples showing that without the boundedness of the covariance, the inequality is not stable. Finally, we study stability in terms of the Wasserstein distance, and show that even for the class of measures with a bounded covariance matrix, it is hopeless to obtain a dimension-free stability result. The counterexamples provided motivate us to put forth a new notion of stability, in terms of proximity to mixtures of the Gaussian distribution. We prove new estimates (some dimension-free) based on this notion. These estimates are strictly stronger than some of the existing stability results in terms of the Wasserstein metric. Our proof techniques rely heavily on stochastic methods.en
dc.publisher.cityParisen
dc.identifier.citationpages16en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris-Dauphineen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.identifier.citationdate2019-10
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2019-12-18T09:32:30Z
hal.person.labIds10358
hal.person.labIds60$$$66
hal.person.labIds117300


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