Colloque / Séminaire

Karhunen-Loève decomposition of Gaussian measures on Banach spaces.

Par Jean-Charles CROIX, Doctorant en Mathématiques Appliquées à l'environnement - Ecole des Mines de St Etienne

Abstract :
The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the so-called Karhunen-L{\`o} eve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated to Gaussian measures on Banach spaces. In the special case of the standard Wiener measure, this decomposition matches with Paul Lévy's construction of Brownian motion.