Product systems arising from L\'evy processe

TL;DR

This paper explores the structure of product systems derived from Banach space-valued Lévy processes, identifying conditions for spatiality and constructing diverse non-isomorphic systems.

Contribution

It establishes when these product systems are completely spatial and constructs a continuum of non-isomorphic type II∞ systems from jump Lévy processes.

Findings

Gaussian Lévy processes with non-degenerate covariance produce type I product systems.

Conditions for product system spatiality are characterized.

A continuum of non-isomorphic type II∞ systems is constructed from jump Lévy processes.

Abstract

This paper investigates the structure of product systems of Hilbert spaces derived from Banach space-valued L\'evy processes. We establish conditions under which these product systems are completely spatial and show that Gaussian L\'evy processes with non-degenerate covariance always give rise to product systems of type I. Furthermore, we construct a continuum of non-isomorphic product systems of type \(\rm{II}\sb\infty\) from pure jump L\'evy processes.