Pitman transforms and Brownian motion in the interval viewed as an affine alcove
Probability
2020-06-11 v4 Representation Theory
Abstract
Pitman's theorem states that if {Bt, t 0} is a one-dimensional Brownian motion, then {Bt -- 2 inf st Bs, t 0} is a three dimensional Bessel process, i.e. a Brownian motion conditioned in Doob sense to remain forever positive. In this paper one gives a similar representation for the Brownian motion in an interval. Due to the double barrier, it is much more involved and only asymptotic. This uses the fact that the interval is an alcove of the Affine Lie algebra A 1 1 .
Cite
@article{arxiv.1808.09182,
title = {Pitman transforms and Brownian motion in the interval viewed as an affine alcove},
author = {Philippe Bougerol and Manon Defosseux},
journal= {arXiv preprint arXiv:1808.09182},
year = {2020}
}