Random walk on a building of type $\tilde{A}_r$ and Brownian motion of the Weyl chamber

Details Random walk on a building of type $\tilde{A}_r$ and Brownian motion of the Weyl chamber Random walk on a building of type $\tilde{A}_r$ and Brownian motion of the Weyl chamber

2006-11-17

arxiv.org/abs/math/0611529v4

Mathematics

Probability

proof of Proposition 6.1 corrected, other minor corrections, to appear in Annales IHP (B)

Scientific paper

In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. 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 \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in \cite{AST}.

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