  This paper presents a multidimensional extension of the Matsumoto-Yor
properties related to exponential functionals of drifted Brownian motion. The
extension involves the interaction of geometric Brownian motions which are
indexed by the vertices of a finite weighted graph, and the random potential
associated with the Vertex Reinforced Jump process on this graph. We prove in
this context a counterpart of Lamperti's transformation, of the Markov property
of the Matsumoto-Yor process and of the intertwining relation.
