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, Volume 45 (2009) no. 2, p. 289-301

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).

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).

DOI : https://doi.org/10.1214/07-AIHP163
Classification:  05C25,  60B10,  60B15,  60C05,  60J10,  60J25,  60J35,  60J60
Keywords: random walk, affine building, root systems, GUE process
@article{AIHPB_2009__45_2_289_0,
     author = {Schapira, Bruno},
     title = {Random walk on a building of type $\~A\_r$ and brownian motion of the Weyl chamber},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {2},
     year = {2009},
     pages = {289-301},
     doi = {10.1214/07-AIHP163},
     zbl = {1218.60003},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_2_289_0}
}
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, Volume 45 (2009) no. 2, pp. 289-301. doi : 10.1214/07-AIHP163. http://www.numdam.org/item/AIHPB_2009__45_2_289_0/

[1] J.-P. Anker, P. Bougerol and T. Jeulin. The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoamericana 18 (2002) 41-97. | MR 1924687 | Zbl 1090.58020

[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. | MR 1302912 | Zbl 0812.60067

[4] P. Biane. Quantum random walk on the dual of SU(n). Probab. Theory Related Fields 89 (1991) 117-129. | MR 1109477 | Zbl 0746.46058

[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. | MR 1186654 | Zbl 0787.60089

[6] P. Biane. Équation de Choquet-Deny sur le dual d'un groupe compact. Probab. Theory Related Fields 94 (1992) 39-51. | MR 1189084 | Zbl 0766.46044

[7] P. Biane, P. Bougerol and N. O'Connell. Littelmann paths and Brownian paths. Duke Math. J. 130 (2005) 127-167. | MR 2176549 | Zbl 1161.60330

[8] P. Billingsley. Convergence of Probability Measures. Wiley Ser. Probab. Statist., Wiley-Intersci. Publ. Wiley, New York, 1999. | MR 1700749 | Zbl 0944.60003

[9] P. Bougerol and T. Jeulin. Brownian bridge on Riemannian symmetric spaces. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 785-790. | MR 1868954 | Zbl 1003.60073

[10] P. Bougerol and T. Jeulin. Brownian bridge on hyperbolic spaces and on homogeneous trees. Probab. Theory Related Fields 115 (1999) 95-120. | MR 1715541 | Zbl 0947.58032

[11] N. Bourbaki. Groupes et algèbres de Lie. Hermann, Paris, 1968; Ch. 4-6. Masson, Paris, 1981. | Zbl 0483.22001

[12] D. I. Cartwright. Spherical harmonic analysis on buildings of type Ãn. Monatsh. Math. 133 (2001) 93-109. | MR 1860293 | Zbl 1008.51019

[13] D. I. Cartwright and W. Woess. Isotropic random walks in a building of type Ãd. Math. Z. 247 (2004) 101-135. | MR 2054522 | Zbl 1060.60070

[14] N. Ethier and G. Kurtz. Markov Processes. Characterization and Convergence. Wiley Ser. Probab. Math. Statist. Wiley, New York, 1986. | MR 838085 | Zbl 0592.60049

[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. | MR 2136870 | Zbl 1087.60058

[16] S. Helgason. Groups and Geometric Analysis. Academic Press, 1984. | MR 754767 | Zbl 0543.58001

[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. | MR 435301 | Zbl 0302.43018

[18] I. G. Macdonald. Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45 (2000), Article B45a. | MR 1817334 | Zbl 1032.33010

[19] N. O'Connell. Random matrices, non-collinding processes and queues. Sém. Probab. XXXVI, LNM 1801. Springer (2003) 165-182. | Numdam | MR 1971584 | Zbl 1041.15019

[20] J. Parkinson. Spherical harmonic analysis on affine buildings. Math. Z. 253 (2006) 571-606. | MR 2221087 | Zbl 1171.43009

[21] J. Parkinson. Buildings and Hecke Algebras. PhD thesis, University of Sydney, 2005. | MR 2206366 | Zbl 1095.20003

[22] J. Parkinson. Isotropic random walks on affine buildings. Ann. Inst. Fourier 57 (2007) 379-419. | Numdam | MR 2310945 | Zbl 1177.60046

[23] M. Ronan. Lectures on Buildings. Perspect. Math. 7. Academic Press, Boston, MA, 1989. | MR 1005533 | Zbl 0694.51001

[24] Br. Schapira. The Heckman-Opdam Markov processes. Probab. Theory Related Fields 138 (2007) 495-519. | MR 2299717 | Zbl 1123.58022