Uses of non-Fock quantum Brownian motion and a quantum martingale representation theorem. (English) Zbl 0569.60055

Hudson, R. L.; Lindsay, J. M.

Quantum probability and applications II, Proc. 2nd Workshop, Heidelberg/Ger. 1984, Lect. Notes Math. 1136, 276-305 (1985).

[For the entire collection see Zbl 0566.00017.]
After reviewing theories of stochastic integration against Fock and non- Fock quantum Brownian motion, we prove a martingale representation theorem for the latter, extending the main result of the authors’ [Stochastic integration and a martingale representation theorem for non- Fock quantum Brownian motion. J. Funct. Anal., to appear] by incorporating an initial space. We construct unitary processes adapted to the filtration of non-Fock quantum Brownian motion and use the martingale representation theorem to characterise such processes in terms of covariantly adapted unitary evolutions [the first author, P. D. F. Ion and K. R. Parthasarathy, Commun. Math. Phys. 83, 261-280 (1982; Zbl 0485.46038)] with a continuity property. The classical limits of the quantum dynamical semigroups associated with these processes are contrasted with those arising in the Fock case.

MSC:

60H05	Stochastic integrals
47D03	Groups and semigroups of linear operators
81P20	Stochastic mechanics (including stochastic electrodynamics)

Keywords:

quantum Brownian motion; martingale representation theorem; quantum dynamical semigroups

Citations:

Zbl 0566.00017; Zbl 0485.46038