Product systems associated to compound Poisson Processes

TL;DR

This paper constructs product systems from stationary Poisson and compound Poisson processes, demonstrating that the associated $E_0$-semigroups are CCR flows, thus linking stochastic processes with operator algebra structures.

Contribution

It introduces a novel connection between product systems derived from Poisson processes and CCR flows, expanding the understanding of their operator algebraic properties.

Findings

Product systems from Poisson processes are associated with CCR flows.

The constructed $E_0$-semigroups are shown to be CCR flows.

Establishes a link between random measures and operator algebra structures.

Abstract

In this paper, we consider a simple test case of multiparameter product systems that arise out of random measures. We associate a product system to a stationary Poisson process and a stationary compound Poisson process. We show that the resulting $E_0$-semigroups are CCR flows.