Classical Versions of Quantum Stochastic Processes Associated with the Oscillator Algebra

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Year:
2015
Type of Publication:
Article
Keywords:
Adapted oscillator algebra, Fock space representation, Operators processes, Levy process, Infinitely divisible laws
Authors:
Habib Rebei; Bader Al-Mohaimeed; Anis Riahi
Journal:
IJISM
Volume:
3
Number:
5
Pages:
245-253
Month:
September
Abstract:
It has been known for a long time that any infinitely divisible distribution (I.D.D) can be realized on a symmetric Fock space with an appropriate noise space. This realization led to a kind of correspondence between Lie algebras and I.D.D. Namely, each I.D.D (or equivalently Lévy process) leads to a such Lie algebra commutation relations. In this context, it was shown (see [1]) that the the quantum stochastic processes corresponding to the bounded form of the oscillator algebra can not cover a large classes of Lévy processes, in particular the non standard Meixner classes. For this reason, we consider the unbounded form of the oscillator algebra called the adapted oscillator algebra. Then, we prove that its Fock representation can give rise to the infinitely divisible processes such as the Gamma, Pascal and the Meixner-Pollaczek.

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