Random walk on a building of type à r and brownian motion of the Weyl chamber
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 289-301.

Dans cet article nous étudions une marche aléatoire sur un immeuble affine de type Ãr, dont la partie radiale renormalisée, converge vers le mouvement Brownien de la chambre de Weyl. Cela fournit une nouvelle discrétisation de ce processus, alternative à celle de Biane (Probab. Theory Related Fields 89 (1991) 117-129). En même temps cela étend en rang supérieur la correspondance à un niveau probabiliste entre les espaces symétriques riemanniens de type non compact et leur version discrète, les immeubles affines, qui fut mise en évidence par Bougerol et Jeulin en rang 1 (C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 785-790). Les principaux ingrédients de la preuve sont une formule combinatoire sur l'immeuble et les estimations du noyau de transition démontrées dans Anker et al. (2006).

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).

DOI : 10.1214/07-AIHP163
Classification : 05C25, 60B10, 60B15, 60C05, 60J10, 60J25, 60J35, 60J60
Mots-clés : random walk, affine building, root systems, GUE process 
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Schapira, Bruno. Random walk on a building of type $Ã_r$ and brownian motion of the Weyl chamber. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 289-301. doi : 10.1214/07-AIHP163. http://www.numdam.org/articles/10.1214/07-AIHP163/

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