Pitman transforms and Brownian motion in the interval viewed as an affine alcove

TL;DR

This paper extends Pitman's theorem to Brownian motion confined in an interval viewed as an affine alcove, providing an asymptotic representation involving affine Lie algebra structures.

Contribution

It introduces a novel representation for Brownian motion in an interval as an affine alcove, generalizing Pitman's theorem using affine Lie algebra concepts.

Findings

Asymptotic representation of Brownian motion in an interval

Connection between Brownian motion and affine Lie algebra alcoves

Extension of Pitman's theorem to double barrier case

Abstract

Pitman's theorem states that if {Bt, t $\ge$ 0} is a one-dimensional Brownian motion, then {Bt -- 2 inf s$\le$t Bs, t $\ge$ 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 .