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

Meyer, P. A.

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 In chapter IX, based on an unpublished paper by Y. Le Jan, multiple stochastic integrals with respect to complex Brownian motion and various products - Wiener, Wick, Grassmann, antisymmetric Wiener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

MSC:

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

Keywords:

Wick product; Grassmann product; multiple stochastic integrals; Markov fields; Wiener products; oscillator Fock space

Citations:

Zbl 0635.00013; Zbl 0633.60074