Hypergroups and Quantum Bessel Processes of Non-integer Dimensions

It is demonstrated how to use a 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(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {BES}(\delta )$$\end{document} are analogously constructed with the aid of the Bessel–Kingman hypergroups for all, not necessarily integer, dimensions δ≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \ge 1$$\end{document}.