A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces

Abstract

This paper deals with a generalization of a result due to Brascamp and Lieb which states that in the space of probabilities with log-concave density with respect to a Gaussian measure on this Gaussian measure is the one which has strongest moments. We show that this theorem remains true if we replace x α by a general convex function. Then, we deduce a correlation inequality for convex functions quite better than the one already known. Finally, we prove results concerning stochastic analysis on abstract Wiener spaces through the notion of approximate limit.