Hypergroups and Quantum Bessel Processes of Non-integer Dimensions

TL;DR

This paper introduces a universal hypergroup-based framework to construct quantum Bessel processes of all dimensions greater than or equal to one, extending classical Bessel process constructions to non-integer dimensions.

Contribution

It presents a novel hypergroup approach that unifies the construction of quantum and classical Bessel processes across all relevant dimensions.

Findings

Constructs quantum Bessel processes for all dimensions ≥ 1 using hypergroups.

Extends classical Bessel process construction to non-integer dimensions.

Provides a universal framework linking hypergroups and stochastic processes.

Abstract

It is demonstrated how to use certain family of commutative hypergroups to provide a universal construction of Biane's quantum Bessel processes of all dimensions not smaller than 1. The classical Bessel processes BES$(\delta)$ are analogously constructed with the aid of the Bessel-Kingman hypergroups for all, not necessarily integer, dimensions $\delta\ge1$.