Hypergroups and Quantum Bessel Processes of Non-integer Dimensions

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 $\delta \ge 1$.