On a gateway between continuous and discrete Bessel and Laguerre processes

TL;DR

This paper explores a deep connection between continuous squared Bessel processes and discrete birth-death processes, revealing bidirectional relationships and inherited properties, which enhance understanding of their interplay and enable information transfer between their semi-groups.

Contribution

It establishes a two-way intertwining relationship between squared Bessel and birth-death processes, revealing new properties and extending the gateway concept to Laguerre semi-groups.

Findings

Identified a bidirectional intertwining relationship.

Discovered inherited properties like discrete self-similarity.

Extended the gateway identity to Laguerre semi-groups.

Abstract

By providing instances of approximation of linear diffusions by birth-death processes, Feller [13], has offered an original path from the discrete world to the continuous one. In this paper, by identifying an intertwining relationship between squared Bessel processes and some linear birth-death processes, we show that this connection is in fact more intimate and goes in the two directions. As by-products, we identify some properties enjoyed by the birth-death family that are inherited from squared Bessel processes. For instance, these include a discrete self-similarity property and a discrete analogue of the beta-gamma algebra. We proceed by explaining that the same gateway identity also holds for the corresponding ergodic Laguerre semi-groups. It follows again that the continuous and discrete versions are more closely related than thought before, and this enables to pass information from one semi-group to the other one.