Beta-gamma algebra identities and Lie-theoretic exponential functionals of Brownian motion

TL;DR

This paper computes the exit law of a hypoelliptic Brownian motion on a solvable Lie group, revealing new identities between gamma variables linked to braid relations and connecting probabilistic and algebraic structures.

Contribution

It introduces explicit formulas for the exit law of a hypoelliptic Brownian motion, uncovering new gamma identities related to Lie algebra braid relations and Lusztig's canonical basis.

Findings

Identifies new gamma variable identities from braid relations

Connects probabilistic laws to algebraic structures in Lie theory

Provides a conditional representation theorem for hypoelliptic Brownian motion

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.