Resumen de Extension du théorème de Cameron--Martin aux translations aléatoires

Xavier Fernique

• Let G be a Gaussian vector taking its values in a separable Fréchet space E. We denote by γ its law and by (H,∥⋅∥) its reproducing Hilbert space. Moreover, let X be an E-valued random vector of law μ. In the first section, we prove that if μ is absolutely continuous relative to γ, then there exist necessarily a Gaussian vector G′ of 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 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 μ being absolutely continuous relative to γ.