Some remarks about the positivity of random variables on a Gaussian probability space

Denis Feyel; A. Suleyman Üstünel

doi:10.1016/j.crma.2004.10.014

Probability Theory

Some remarks about the positivity of random variables on a Gaussian probability space
[Quelques remarques sur la positivité des variables aléatoires définies sur un espace gaussien.]

Présenté par : Paul Malliavin

Denis Feyel ¹ ; A. Suleyman Üstünel ²

¹ Université d'Evry-Val-d'Essone, département de mathématiques, 91025 Evry cedex, France
² ENST, département Infres, 46, rue Barrault, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 873-877.

Résumé
Abstract

Soit $(W, H, μ)$ un espace de Wiener abstrait et soit $L \in L \log L$ une variable aléatoire positive. A l'aide de la théorie de transport de mesure de Monge–Kantorovitch, nous montrons que le noyau de la projection de L dans le second chaos de Wiener est un opérateur de spectre inférieurement borné et que l'opérateur correspondant est inférieurement borné par un opérateur Hilbert–Schmidt semi-positif.

Let $(W, H, μ)$ be an abstract Wiener space and let $L \in L \log L (μ)$ is a positive random variable. Using the measure transportation of Monge–Kantorovitch, we prove that the operator corresponding to the kernel of the projection of L on the second Wiener chaos is lower bounded by a semi-positive Hilbert–Schmidt operator.

Reçu le : 2004-10-05
Accepté le : 2004-10-11
Publié le : 2004-11-13

DOI : 10.1016/j.crma.2004.10.014

Affiliations des auteurs :

Denis Feyel ¹ ; A. Suleyman Üstünel ²

¹ Université d'Evry-Val-d'Essone, département de mathématiques, 91025 Evry cedex, France
² ENST, département Infres, 46, rue Barrault, 75013 Paris, France

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     title = {Some remarks about the positivity of random variables on a {Gaussian} probability space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {873--877},
     publisher = {Elsevier},
     volume = {339},
     number = {12},
     year = {2004},
     doi = {10.1016/j.crma.2004.10.014},
     language = {en},
}

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Denis Feyel; A. Suleyman Üstünel. Some remarks about the positivity of random variables on a Gaussian probability space. Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 873-877. doi : 10.1016/j.crma.2004.10.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.10.014/

Bibliographie
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