Comparison inequalities on Wiener space

Abstract

We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross-Sobolev space $D^{1,2}$ of random variables with a square-integrable Malliavin derivative, we let $Gamma_{F,G}=$ where $D$ is the Malliavin derivative operator and $L^{-1}$ is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. We use $Γ$ to extend the notion of covariance and canonical metric for vectors and random fields on Wiener space, and prove corresponding non-Gaussian comparison inequalities on Wiener space, which extend the Sudakov-Fernique result on comparison of expected suprema of Gaussian fields, and the Slepian inequality for functionals of Gaussian vectors. These results are proved using a so-called smart-path method on Wiener space, and are illustrated via various examples. We also illustrate the use of the same method by proving a Sherrington-Kirkpatrick universality result for spin systems in correlated and non-stationary non-Gaussian random media.