We explicitly compute the exit law of a certain hypoelliptic Brownian motionon a solvable Lie group. The underlying random variable can be seen as amultidimensional exponential functional of Brownian motion. As a consequence,we obtain hidden identities in law between gamma random variables as theprobabilistic manifestation of braid relations. The classical beta-gammaalgebra 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 ourhypoelliptic Brownian motion conditioned on exiting at a fixed point to acertain deterministic transform of Brownian motion. The identities in law between gamma variables tropicalize to identitiesbetween exponential random variables. These are continuous versions ofidentities between geometric random variables related to changes ofparametrizations in Lusztig's canonical basis. Hence, we see that the exit lawof our hypoelliptic Brownian motion is the geometric analogue of a simplenatural measure on Lusztig's canonical basis.
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