White noise quantum time shifts

In the present paper we extend the notion of quantum time shift, and the related results obtained in \cite{[abo06]}, 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 L\'evy processes much wider than the usual Brownian motion. In particular this class processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical L\'evy processes with a L\'evy 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 L\'evy 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.