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

Philippe Bougerol (1) , Manon Defosseux (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

Pitman's theorem states that if {Bt, t ≥ 0} is a one-dimensional Brownian motion, then {Bt − 2 inf s≤t 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 .

Domaines: Probabilités [math.PR]; Théorie des représentations [math.RT]

Philippe Bougerol, Manon Defosseux. Pitman transforms and Brownian motion in the interval viewed as an affine alcove. 2018. ⟨hal-01861862v1⟩