In this paper we study a random walk on an affine building of type Ãr, whose radial part, when suitably normalized, converges toward the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane (Probab. Theory Related Fields 89 (1991) 117–129). This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one (C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 785–790). The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in Anker et al. (2006).
@article{1241024671,
author = {Schapira, Bruno},
title = {Random walk on a building of type \~A<sub>
r
</sub> and Brownian motion of the Weyl chamber},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {45},
number = {1},
year = {2009},
pages = { 289-301},
language = {en},
url = {http://dml.mathdoc.fr/item/1241024671}
}
Schapira, Bruno. Random walk on a building of type Ã
r
and Brownian motion of the Weyl chamber. Ann. Inst. H. Poincaré Probab. Statist., Tome 45 (2009) no. 1, pp. 289-301. http://gdmltest.u-ga.fr/item/1241024671/