Stokes formula on the Wiener space and n-dimensional Nourdin–Peccati analysis

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.