Electronic Journal of Probability

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

Reda Chhaibi

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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.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 108, 20 pp.

Accepted: 19 October 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60J65: Brownian motion [See also 58J65]

Beta-gamma algebra identities Exponential functionals of Brownian motion Braid relations Total positivity Brownian motion

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Chhaibi, Reda. Beta-gamma algebra identities and Lie-theoretic exponential functionals of Brownian motion. Electron. J. Probab. 20 (2015), paper no. 108, 20 pp. doi:10.1214/EJP.v20-3666. https://projecteuclid.org/euclid.ejp/1465067214

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