Squared white noise and other non-Gaussian noises as Levy processes on real Lie algebras

摘要

It is shown how the relations of the renormalized squared white noise defined udby Accardi, Lu, and Volovich cite{accardi+lu+volovich99} can be realized as udfactorizable current representations or L'evy processes on the real Lie udalgebra $eufrak{sl}_2$. This allows to obtain its It^o table, which turns udout to be infinite-dimensional. The linear white noise without or with number udoperator is shown to be a L'evy process on the Heisenberg-Weyl Lie algebra udor the oscillator Lie algebra. Furthermore, a joint realization of the linear udand quadratic white noise relations is constructed, udbut it is proved that no such realizations exist with a vacuum that is an eigenvector of the central element and the annihilator. Classical L'evy processes are shown to arise as components of L'evy process on real Lie algebras and their distributions are characterized.