  We explicitly compute the exit law of a certain hypoelliptic Brownian motion
on a solvable Lie group. The underlying random variable can be seen as a
multidimensional exponential functional of Brownian motion. As a consequence,
we obtain hidden identities in law between gamma random variables as the
probabilistic manifestation of braid relations. The classical beta-gamma
algebra identity corresponds to the only braid move in a root system of type
$A_2$. The other ones seem new.
  A key ingredient is a conditional representation theorem. It relates our
hypoelliptic Brownian motion conditioned on exiting at a fixed point to a
certain deterministic transform of Brownian motion.
  The identities in law between gamma variables tropicalize to identities
between exponential random variables. These are continuous versions of
identities between geometric random variables related to changes of
parametrizations in Lusztig's canonical basis. Hence, we see that the exit law
of our hypoelliptic Brownian motion is the geometric analogue of a simple
natural measure on Lusztig's canonical basis.
