The Lévy Laplacian and stochastic processes (Q2751525)

The author and \textit{A. H. Tsoi} [in: Quantum information, 159-171 (1999; Zbl 0982.60063)] found a Hilbert space of generalized Gaussian white noise functionals, to which the Lévy Laplacian \(\Delta_L\) can be extended as self-adjoint operator \(\overline{\Delta_L}\). Suppose that \(e^{th(z)}\) is a characteristic function of a homogeneous stochastic process with independent increments. The function \(h(z)\) admits the well-known integral representation. The author shows that the operator \(h(-\overline{\Delta_L})\) is a generator of a Markov process. In particular, the operator \((-\overline{\Delta_L})^\alpha\), \(0<\alpha <2\), can be considered as a generator of an infinite-dimensional stable process. Relations between the above processes and the infinite-dimensional Ornstein-Uhlenbeck process are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].