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 Wesołowski [H. Massam, J. Wesołowski, 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].