Comparison inequalities on Wiener space

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• 作者：Ivan Nourdin ; Giovanni Peccati ; Frederi G. Viens
• 关键词：60F05 ; 60G15 ; 60H05 ; 60H07
• 刊名：Stochastic Processes and their Applications
• 出版年：April, 2014
• 年：2014
• 卷：124
• 期：4
• 页码：1566-1581
• 全文大小：260 K

文摘

We define a covariance-type operator on Wiener space: for and two random variables in the Gross-Sobolev space of random variables with a square-integrable Malliavin derivative, we let , where is the Malliavin derivative operator and is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. We use 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.