Narrow Results

Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

In this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Lévy–Laplacian is obtained as the usual Volterra–Gross Laplacian using the Cesàro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise.

In this paper, we construct a class of infinitely divisible distributions on Gel′fand triple. Based on this construction, we define Lévy processes on Gel′fand triple and give their Lévy–Itô decompositions. Then, we construct the general Lévy white noises on Gel′fand triple. By using the Riemann–Liouville fractional integral method, we define the general fractional Lévy noises on Gel′fand triple and investigate their distribution properties.