Abstract
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.
Highlights
We explicitly compute the exit law of a certain hypoelliptic Brownian motion on a solvable Lie group
We focus on the N part which plays the role of a multidimensional exponential functional of Brownian motion
Ejp.ejpecp.org groups with higher rank, the presented construction gives the explicit law of multiple exponential functionals of Brownian motion
Summary
Let G be a complex semi-simple group of rank r. In the case of the group SL2, one recovers Dufresne’s identity in law on the exponential functional of a Brownian motion with drift [8]. Groups with higher rank, the presented construction gives the explicit law of multiple exponential functionals of Brownian motion. We begin by stating the three main theorems 2.12, 2.16 and 2.21, after the necessary preliminaries on Lie theory and total positivity. It implies the discrete version involving geometric random variables. Thanks to results from [7] and [6], we reduce the problem to an induction whose base case is a result by Matsumoto and Yor on a relationship between Brownian motions with opposite drifts [18].
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