Marche aléatoire sur un immeuble affine de type $\tilde{A}_r$ et mouvement brownien de la chambre de Weyl

Bruno Schapira (1, 2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this paper we study a random walk on an affine building, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber (for the type $A$). This gives a new discrete approximation of this process, which is different from the one of Biane \cite{Bia2}. The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in \cite{AST}. Moreover our result extends in higher rank the correspondence at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously obtained by Bougerol and Jeulin in rank one \cite{BJ}.

Mots clés: Marche aléatoire; immeuble affine; système de racines; processus GUE.; processus GUE

Domaines: Probabilités [math.PR]

Bruno Schapira. Marche aléatoire sur un immeuble affine de type $\tilde{A}_r$ et mouvement brownien de la chambre de Weyl. 2006. ⟨hal-00114681v1⟩