Eléments de probabilités quantiques. IX: Calculs antisymétriques et “supersymétriques” en probabilités. X: Calculs avec des noyaux discrets. (Elements of quantum probabilities. IX: Antisymmetric and supersymmetric calculus. X: Calculus with discrete kernels). (French) Zbl 0659.60077

Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 101-123, 124-128 (1988).
[For the entire collection see Zbl 0635.00013.]
The paper continues the study of algebras of multiple stochastic integrals initiated in the previous chapters [for Chapters VI-VIII see Séminaire de probabilités XXI, Strasbourg/France, ibid. 1247, 34-80 (1987; Zbl 0633.60074)]. In chapter IX, based on an unpublished paper by Y. Le Jan, multiple stochastic integrals with respect to complex Brownian motion and various products - Wiener, Wick, Grassmann, antisymmetric Wiener - are considered. Exponentials in the corresponding algebras are calculated. Applications to the determination of polynomials of Markov fields are discussed.
Chapter X is a short note on the application of Maassen’s kernel calculus to the computation of the Wick and Wiener products and to the normal ordering of operators in the oscillator Fock space.
Reviewer: A.Holevo


60H05 Stochastic integrals
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
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