Quantum stochastic calculus and a boson Levy theorem

Kalyanapuram Rangachari Parthasarathy

R. L. Hudson, Kalyanapuram Rangachari Parthasarathy

Shang Chen, R. L. Hudson

Weconsider the analogueof Levy area, definedas an iterated stochastic integral, obtained by replacing two independent component one- dimensional Brownian motions by the mutually non-commuting momen- tum and position Brownian motions P and Q of either Fock or non-Fock quantum stochastic calculus, which are also stochastically independent in a certain sense. We show that the resulting quantum Levy area is trivially distributed in the Fock case, but has a non-trivial distribution in non-Fock quantum stochastic calculus which, after rescaling, interpolates between the trivial distribution and that of classical Levy area in the infinite tempera- ture limit. We also show that it behaves differently from the classical Levy area under a kind of time reversal, in both the Fock and non-Fock cases. 2000 AMS Mathematics Subject Classification: Primary: 81S25.

Timothy M. W. Eyre

Omar Alzeley, Aymen Ettaieb, Hafedh M. S. Rguigui

ABSTRACT We establish the commutation relations of the quantum Lévy operators, consisting of the generalized Lévy Gross Laplacian and its adjoint, annihilation operator, creation operator, and the Lévy number operator. In particular, such quantum Lévy operators span a nonsolvable ‐Lie algebra. We then give a generalization of the Hudson–Parthasarathy stochastic integrals in terms of the quantum Lévy operators and derive a noncommutative Itô formula.

Александр Семенович Холево