Some remarks about the positivity of random variables on a Gaussian   probability space
  Let $(W,H,\mu)$ be an abstract Wiener space and $L$ be a probability density of class LlogL. Using the measure transportation of Monge-Kantorovitch, we prove that the kernel of the projection of L on the second Wiener chaos defines an (Hilbert-Schmidt) operator which is lower bounded by another Hilbert-Schmidt operator. 
D. Feyel and A.S. Ustunel