Extension du théorème de Cameron-Martin aux translations aléatoires. II. Intégrabilité des densités

Abstract

Let Gbe a Gaussian vector on a probability space (0, F, P) taking its values in a separable Frechet space E. We denote by (its law and by $(H,\parallel \cdot {{\parallel }_{H}})$ its reproducing kernel Hilbert space. Let moreover X be an Evalued random vector of law µ.

We know 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.

In this work, we will compare the γ-integrability of the density D = dµ/dγ and the P-integrability of ‖Z‖ _H .