  Consider a negatively drifted one dimensional Brownian motion starting at
positive initial position, its first hitting time to 0 has the inverse Gaussian
law. Moreover, conditionally on this hitting time, the Brownian motion up to
that time has the law of a 3- dimensional Bessel bridge. In this paper, we give
a generalization of this result to a family of Brownian motions with
interacting drifts, indexed by the vertices of a conductance network. The
hitting times are equal in law to the inverse of a random potential that
appears in the analysis of a self-interacting process called the Vertex
Reinforced Jump Process ([17, 18]). These Brownian motions with interacting
drifts have remarkable properties with respect to restriction and conditioning,
showing hidden Markov properties. This family of processes are closely related
to the martingale that plays a crucial role in the analysis of the vertex
reinforced jump process and edge reinforced random walk ([18]) on infinite
graphs.
