Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis |
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Authors: | Hé lè ne Airault,Paul Malliavin |
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Affiliation: | a Université de Picardie Jules Verne, Insset, 48 rue Raspail, 02100 Saint-Quentin (Aisne), LAMFA, UMR 6140, 33, rue Saint-Leu, 80000 Amiens, France b 10, rue Saint-Louis-en-l'Ile, 75004 Paris, France c Dept. Statistics and Dept. Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067, USA |
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Abstract: | Extensions of the Nourdin-Peccati analysis to Rn-valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein's lemma), to classical analysis on Wiener space. Partial differential equations for the density of an Rn-valued centered random variable Z=(Z1,…,Zn) are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix (−DL−1Zi|DZj), i,j=1,2,…,n, where D and L are respectively the derivative operator and the generator of the Ornstein-Uhlenbeck semigroup on Wiener space. |
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Keywords: | Wiener space Quasi-sure analysis Density formula Nourdin-Peccati analysis Stein's lemma Conditional probability |
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