2014/04/30 by Reda Chhaibi
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Braid group #Brownian excursion #Brownian motion #Differential equation #Diffusion process #Geometric Brownian motion #Hypoelliptic operator #Mathematical analysis #Mathematics #Pure mathematics #Random Matrices and Applications #math.GR #math.PR #msc:60B15 #msc:60B20 #msc:60J65
paper · pdf · doi:10.1214/ejp.v20-3666
published as Electron. J. Probab. 20 (2015), no. 108, pages 1-20 · 32 pages, 1 appendix; this paper extends and replaces the subsections 6.3 and 6.4 from the author's phD thesis, available as arXiv:1302.0902; v3: Published version with an additional review section
openalex publication_date 2015/01/01 · arxiv created 2016/04/27 · arxiv updated 2016/04/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 A2. 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.