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

Pitman's theorem states that if {Bt, t ≥ 0} is a one-dimensional Brownian motion, then {Bt − 2 inf s≤t Bs, three dimensional Bessel process, i.e. motion conditioned in Doob sense to remain forever positive. In this paper one gives similar representation for the an interval. Due double barrier, it much more involved and only asymptotic. This uses fact interval alcove of Affine Lie algebra A 1 .