Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 987-1019.

Dans cet article, nous démontrons qu’un mouvement brownien sur un arbre aléatoire continu est en fait la limite rééchelonnée d’un certain type de marches aléatoires simples; ces marches aléatoires simples évoluent sur n’importe quelle famille de graphes d’arbres discrets ordonnés de n sommets, dont les fonctions de recherche en profondeur convergent vers une excursion brownienne lorsque n. Nous prouvons deux versions de notre résultat principal: une première conditionnelle sur les réalisations typiques des arbres, ainsi qu’une seconde où l’on prend la moyenne sur toutes les réalisations des arbres. Les hypothèses de cet article couvrent l’exemple important d’une marche aléatoire simple sur les arbres générés par le processus de branchement de Galton-Watson, étant donné la taille de la population totale.

In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete n-vertex ordered graph trees whose search-depth functions converge to the brownian excursion as n. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total population size.

DOI : 10.1214/07-AIHP153
Classification : 60K37, 60G99, 60J15, 60J80, 60K35
Mots clés : continuum random tree, brownian motion, random graph tree, random walk, scaling limit
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Croydon, David. Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 987-1019. doi : 10.1214/07-AIHP153. http://www.numdam.org/articles/10.1214/07-AIHP153/

[1] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1991) 1-28. | MR | Zbl

[2] D. Aldous. The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990) 23-70. London Math. Soc. Lecture Note Ser. 167. Cambridge Univ. Press, 1991. | MR | Zbl

[3] D. Aldous. The continuum random tree. III. Ann. Probab. 21 (1993) 248-289. | MR | Zbl

[4] M. T. Barlow. Diffusions on fractals. Lectures on Probability Theory and Statistics (Saint-Flour, 1995) 1-121. Lecture Notes in Math. 1690. Springer, Berlin, 1998. | MR | Zbl

[5] M. T. Barlow and T. Kumagai. Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 (2006) 33-65 (electronic). | MR | Zbl

[6] P. Billingsley. Probability and Measure, 3rd edition. Wiley, New York, 1995. | MR | Zbl

[7] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | MR | Zbl

[8] A. N. Borodin. The asymptotic behavior of local times of recurrent random walks with finite variance. Teor. Veroyatnost. i Primenen. 26 (1981) 769-783. | MR | Zbl

[9] D. A. Croydon. Volume growth and heat kernel estimates for the continuum random tree. Probab. Theory Related Fields. To appear. DOI: 10.1007/S00440-007-0063-4. | MR | Zbl

[10] R. M. Dudley. Sample functions of the Gaussian process. Ann. Probab. 1 (1973) 66-103. | MR | Zbl

[11] T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553-603. | MR | Zbl

[12] S. N. Evans, J. Pitman and A. Winter. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 (2006) 81-126. | MR | Zbl

[13] M. Fukushima, Y. Ōshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co., Berlin, 1994. | MR | Zbl

[14] A. Greven, P. Pfaffelhuber and A. Winter. Convergence in distribution of random metric measure spaces (λ-coalescent measure trees). Preprint. Available at http://arxiv.org/abs/math/0609801. | MR

[15] T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41 (2000) 1244-1293. | MR | Zbl

[16] S. Janson and J.-F. Marckert. Convergence of discrete snakes. J. Theoret. Probab. 18 (2005) 615-647. | MR | Zbl

[17] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, New York, 2002. | MR | Zbl

[18] H. Kesten. Sub-diffusive behavior of random walk on a random cluster. Unpublished proof.

[19] H. Kesten. Sub-diffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 425-487. | Numdam | MR | Zbl

[20] J. Kigami. Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 (1995) 48-86. | MR | Zbl

[21] J. Kigami. Analysis on Fractals. Cambridge Univ. Press, 2001. | MR | Zbl

[22] W. B. Krebs. Brownian motion on the continuum tree. Probab. Theory Related Fields 101 (1995) 421-433. | MR | Zbl

[23] T. Kumagai. Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci. 40 (2004) 793-818. | MR | Zbl

[24] M. B. Marcus and J. Rosen. Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab. 20 (1992) 1603-1684. | MR | Zbl

[25] P. Révész. Local time and invariance. Analytical Methods in Probability Theory (Oberwolfach, 1980) 128-145. Lecture Notes in Math. 861. Springer, Berlin, 1981. | MR | Zbl

[26] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. | MR | Zbl

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