  A classic result on the 1-dimensional Brownian motion shows that
conditionally on its first hitting time of 0, it has the distribution of a
3-dimensional Bessel bridge. By applying a certain time-change to this result,
Matsumoto and Yor showed a theorem giving a relation between Brownian motions
with opposite drifts. The relevant time change is the one appearing in
Lamperti's relation.
  Sabot and Zeng showed that a family of Brownian motions with interacting
drifts, conditioned on the vector of hitting times of 0, also has the
distribution of independent 3-dimensional Bessel bridges. Moreover, the
distribution of these hitting times is related to a random potential that
appears in the study of the vertex-reinforced jump process.
  The aim of this paper is to prove a multivariate version of the Matsumoto-Yor
opposite drift theorem, by applying a Lamperti-type time change to the previous
family of interacting Brownian motions. Difficulties arise since the time
change progresses at different speeds on different coordinates.
