Élements de probabilités quantiques. VI. (Elements of quantum probability. VI). (French) Zbl 0633.60074

Meyer, P. A.

Sémin. probabilités XXI, Lect. Notes Math. 1247, 34-80 (1987).

[For the entire collection see Zbl 0606.00022.]
This is a continuation of the author’s seminar notes on Quantum probability, parts I-V, published in Probabilités XX, Lect. Notes Math. 1204, 186-313 (1986; Zbl 0604.60001). The paper consists of three parts.
Part VI is devoted to a revision of the notion of stochastic integrals based on quantum stochastic calculus, discussion of several products of stochastic integrals (Wiener, Poisson, Clifford products), and embedding of infinitely divisible random variables in the Fock space.
In Part VII, the “non-Fock” representation of canonical commutation relations, namely, those corresponding to universally invariant quasi- free (Gaussian) states and the corresponding stochastic calculus are surveyed.
Part VIII contains a discussion of stopping times for quantum stochastic processes in Fock space, following the work of K. R. Parthasarathy and K. B. Sinha, Probab. Theory Relat. Fields 75, 431-458 (1987; Zbl 0599.60044). Strong Markov property of the Fock space, namely, factorization with respect to arbitrary stopping times is demonstrated.

Reviewer: A.S.Holevo

MSC:

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

Keywords:

Quantum probability; notion of stochastic integrals based on quantum stochastic calculus; products of stochastic integrals; commutation relations; Fock space; Strong Markov property of the Fock space

Citations:

Zbl 0606.00022; Zbl 0604.60001; Zbl 0599.60044