Extension du théorème de Cameron–Martin aux translations aléatoires

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.