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Comptes Rendus Mathématique
Volume 335, n° 1
pages 65-68 (2002)
Doi : S1631-073X(02)02434-2
Received : 5 Mars 2002 ; 
Extension du théorème de Cameron-Martin aux translations aléatoires
Extension of the Cameron-Martin theorem to random translations
 

Xavier Fernique
Institut de recherche mathématique avancée, Université Louis Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg cédex, France 

Note présentée par Marc Yor

Résumé

Soit γ une probabilité gaussienne (centrée) sur un espace de Fréchet séparable et localement convexe E ; soit (H ,‖·‖) l'espace auto-reproduisant associé. On montre que si une probabilité μ sur E est absolument continue relativement à γ , alors il existe un vecteur aléatoire G de loi γ et un vecteur aléatoire Z à valeurs dans H tel que G +Z ait la loi μ ; on utilise pour cela les inégalités isopérimétriques gaussiennes. On montre ensuite que dans certaines situations une telle condition, nécessaire pour l'absolue continuité, est aussi suffisante ; on utilise pour cela le théorème classique de Cameron-Martin et les propriétés d'invariance des probabilités gaussiennes par rotation. Pour citer cet article : X. Fernique, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 65-68.

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Abstract

Let G be a Gaussian vector taking its values in a locally convex separable Fréchet space E . We denote by γ its law and by (H ,‖·‖) its reproducing Hilbert space. Let moreover X be an E -valued random vector of law μ . We prove that if μ is absolutely continuous relatively to γ , then there exist necessarly a Gaussian vector G ′ of the law γ and an H -valued random vector Z such that G ′+Z has the law μ of X . This fact is a direct consequence of isoperimetric properties of Gaussian vector. We show that in many situations, such condition is sufficient for μ being absolutely continuous relatively to γ , using classical Cameron-Martin theorem and invariance properties of Gaussian measures. To cite this article: X. Fernique, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 65-68.

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