Hitting times of Brownian motion and the Matsumoto–Yor property on trees

The Matsumoto–Yor property in the bivariate case was originally defined through properties of functionals of the geometric Brownian motion. A multivariate version of this property was described in the language of directed trees and outside of the framework of stochastic processes in Massam and Weso?owski H. Massam, J. Weso?owski, The Matsumoto–Yor property on trees, Bernoulli 10 (2004) 685–700]. Here we propose its interpretation through properties of hitting times of Brownian motion, extending the interpretation given in the bivariate case in Matsumoto and Yor H. Matsumoto, M. Yor, Interpretation via Brownian motion of some independence properties between GIG and gamma variables, Statist. Probab. Lett. 61 (2003) 253–259].