Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Random walk on a building of type Ãr and Brownian motion of the Weyl chamber

Bruno Schapira

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Abstract

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).

Résumé

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).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 2 (2009), 289-301.

Dates
First available in Project Euclid: 29 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1241024671

Digital Object Identifier
doi:10.1214/07-AIHP163

Mathematical Reviews number (MathSciNet)
MR2521404

Zentralblatt MATH identifier
1218.60003

Subjects
Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 60B10: Convergence of probability measures 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65]

Keywords
Random walk Affine building Root systems GUE process

Citation

Schapira, Bruno. Random walk on a building of type à r and Brownian motion of the Weyl chamber. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 2, 289--301. doi:10.1214/07-AIHP163. https://projecteuclid.org/euclid.aihp/1241024671


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References

  • [1] J.-P. Anker, P. Bougerol and T. Jeulin. The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoamericana 18 (2002) 41–97.
  • [2] J.-P. Anker, Br. Schapira and B. Trojan. Heat kernel and Green’s function estimates on affine buildings of type Ãr. arXiv:math/0612385, 2006.
  • [3] P. Biane. Quelques propriétés du mouvement Brownien dans un cône. Stochastic Process. Appl. 53 (1994) 233–240.
  • [4] P. Biane. Quantum random walk on the dual of SU(n). Probab. Theory Related Fields 89 (1991) 117–129.
  • [5] P. Biane. Minuscule weights and random walks on lattices. In Quantum Probability and Related Topics, QP-PQ, VII. World Sci. Publishing, River Edge, NJ, 1992, pp. 51–65.
  • [6] P. Biane. Équation de Choquet-Deny sur le dual d’un groupe compact. Probab. Theory Related Fields 94 (1992) 39–51.
  • [7] P. Biane, P. Bougerol and N. O’Connell. Littelmann paths and Brownian paths. Duke Math. J. 130 (2005) 127–167.
  • [8] P. Billingsley. Convergence of Probability Measures. Wiley Ser. Probab. Statist., Wiley-Intersci. Publ. Wiley, New York, 1999.
  • [9] P. Bougerol and T. Jeulin. Brownian bridge on Riemannian symmetric spaces. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 785–790.
  • [10] P. Bougerol and T. Jeulin. Brownian bridge on hyperbolic spaces and on homogeneous trees. Probab. Theory Related Fields 115 (1999) 95–120.
  • [11] N. Bourbaki. Groupes et algèbres de Lie. Hermann, Paris, 1968; Ch. 4–6. Masson, Paris, 1981.
  • [12] D. I. Cartwright. Spherical harmonic analysis on buildings of type Ãn. Monatsh. Math. 133 (2001) 93–109.
  • [13] D. I. Cartwright and W. Woess. Isotropic random walks in a building of type Ãd. Math. Z. 247 (2004) 101–135.
  • [14] N. Ethier and G. Kurtz. Markov Processes. Characterization and Convergence. Wiley Ser. Probab. Math. Statist. Wiley, New York, 1986.
  • [15] L. Gallardo and M. Yor. Some new examples of Markov processes which enjoy the time-inversion property. Probab. Theory Related Fields 132 (2005) 150–162.
  • [16] S. Helgason. Groups and Geometric Analysis. Academic Press, 1984.
  • [17] I. G. Macdonald. Spherical Functions on a Group of p-adic Type, Vol. 2. Publications of the Ramanujan Institute, Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, 1971.
  • [18] I. G. Macdonald. Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45 (2000), Article B45a.
  • [19] N. O’Connell. Random matrices, non-collinding processes and queues. Sém. Probab. XXXVI, LNM 1801. Springer (2003) 165–182.
  • [20] J. Parkinson. Spherical harmonic analysis on affine buildings. Math. Z. 253 (2006) 571–606.
  • [21] J. Parkinson. Buildings and Hecke Algebras. PhD thesis, University of Sydney, 2005.
  • [22] J. Parkinson. Isotropic random walks on affine buildings. Ann. Inst. Fourier 57 (2007) 379–419.
  • [23] M. Ronan. Lectures on Buildings. Perspect. Math. 7. Academic Press, Boston, MA, 1989.
  • [24] Br. Schapira. The Heckman–Opdam Markov processes. Probab. Theory Related Fields 138 (2007) 495–519.