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Communications in Mathematical Physics

Renormalized Squares of White Noise¶and Other Non-Gaussian Noises as Lévy Processes¶on Real Lie Algebras

Luigi Accardi, Uwe Franz, Michael Skeide

Published Jun 1, 2002• 95 citations
Publisher
DOI
10.1007/s002200200647

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Abstract In the present paper, we extend the notion of quantum time shift, and the related results obtained in [Accardi, Barhoumi and Ouerdiane, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9: 215–248, 2006], from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brownian motion. In particular, this class of processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical Lévy processes with a Lévy measure with finite variance. Finally, we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of Lévy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way comes somehow as a surprise.
Quantum time shifts are a very special class of continuous time random walk type additive processes on Lie algebras whose exponentiation gives rise to quantum Markov processes whose infinitesimal generators possess a total set of unitary eigen-operators: such a class of quantum Markov generators was not known in the literature.This property generalizes the known fact that trigonometric exponentials are eigenvectors of the usual classical Laplacian and was first noticed in connection with the quantum Laplacian in [5].In that case the eigen-operators turn out to be the Weyl operators, which are known to be non-commutative extensions of the trigonometric exponentials.A generalization of these results, from the Weyl algebra to the harmonic oscillator algebra was obtained in [6].In the present paper we extend the above results to the renormalized square of white noise algebra.This provides among other things, a quantum extension of the Markov generators of the Meixner processes.To our knowledge such an extension was not previously known.