THE QUANTUM DECOMPOSITION ASSOCIATED WITH THE LÉVY
WHITE NOISE PROCESSES WITHOUT MOMENTS
Luigi Accardi
Habib Rebei
Anis Riahi
Abstract: The theory of one-mode type Interacting Fock Space (IFS) allows us to construct the
quantum decomposition associated with stochastic processes on with moments of any
order. The problem to extend this result to processes without moments of any order is still
open but the Araki–Woods–Parthasarathy–Schmidt characterization of Lévy processes in
terms of boson Fock spaces, canonically associated with the Lévy–Khintchine functions
of these processes, provides a quantum decomposition for them which is based
on boson creations, annihilation and preservation operators rather than on their
IFS counterparts. In order to compare the two quantum decompositions in their
common domain of application (i.e., the Lévy processes with moments of all orders)
the first step is to give a precise formulation of the quantum decomposition for
these processes and the analytical conditions of its validity. We show that these
conditions distinguish three different notions of quantum decomposition of a Lévy
process on according to the existence of second or only first moments, or no
moments at all. For the last class a multiplicative renormalization procedure is needed.
2000 AMS Mathematics Subject Classification: Primary: 60J65; Secondary: 60J45,
60H40.
Keywords and phrases: Quantum decomposition, Kolmogorov representation, Lévy
white noise processes, Araki–Woods–Parthasarathy–Schmidt characterization.