Extension du théorème de Cameron--Martin aux translations aléatoires

Abstract

Let G be a Gaussian vector taking its values in a separable Fréchet space E. We denote by $\gamma$ its law and by $(H,\Vert\!\cdot\!\Vert)$ its reproducing Hilbert space. Moreover, let X be an E-valued random vector of law $\mu$. In the first section, we prove that if $\mu$ is absolutely continuous relative to $\gamma$, then there exist necessarily a Gaussian vector $G'$ of law $\gamma$ and an H-valued random vector Z such that $G' + Z$ has the law $\mu$ of X. This fact is a direct consequence of concentration properties of Gaussian vectors and, in some sense, it is an unexpected achievement of a part of the Cameron--Martin theorem.

In the second section, using the classical Cameron--Martin theorem and rotation invariance properties of Gaussian probabilities, we show that, in many situations, such a condition is sufficient for $\mu$ being absolutely continuous relative to $\gamma$.