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, Volume 44 (2008) no. 6, p. 987-1019

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.

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.

DOI : https://doi.org/10.1214/07-AIHP153
Classification:  60K37,  60G99,  60J15,  60J80,  60K35
Keywords: continuum random tree, brownian motion, random graph tree, random walk, scaling limit
@article{AIHPB_2008__44_6_987_0,
     author = {Croydon, David},
     title = {Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {6},
     year = {2008},
     pages = {987-1019},
     doi = {10.1214/07-AIHP153},
     zbl = {1187.60083},
     mrnumber = {2469332},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2008__44_6_987_0}
}
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, Volume 44 (2008) no. 6, pp. 987-1019. doi : 10.1214/07-AIHP153. http://www.numdam.org/item/AIHPB_2008__44_6_987_0/

[1] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1991) 1-28. | MR 1085326 | Zbl 0722.60013

[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 1166406 | Zbl 0791.60008

[3] D. Aldous. The continuum random tree. III. Ann. Probab. 21 (1993) 248-289. | MR 1207226 | Zbl 0791.60009

[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 1668115 | Zbl 0916.60069

[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 2247823 | Zbl 1110.60090

[6] P. Billingsley. Probability and Measure, 3rd edition. Wiley, New York, 1995. | MR 1324786 | Zbl 0822.60002

[7] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | MR 1700749 | Zbl 0944.60003

[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 636771 | Zbl 0474.60056

[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 2357676 | Zbl 1133.62066

[10] R. M. Dudley. Sample functions of the Gaussian process. Ann. Probab. 1 (1973) 66-103. | MR 346884 | Zbl 0261.60033

[11] T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553-603. | MR 2147221 | Zbl 1070.60076

[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 2221786 | Zbl 1086.60050

[13] M. Fukushima, Y. Ōshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co., Berlin, 1994. | MR 1303354 | Zbl 0838.31001

[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 2520129 | Zbl pre05601408

[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 1757958 | Zbl 0977.82022

[16] S. Janson and J.-F. Marckert. Convergence of discrete snakes. J. Theoret. Probab. 18 (2005) 615-647. | MR 2167644 | Zbl 1084.60049

[17] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, New York, 2002. | MR 1876169 | Zbl 0996.60001

[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 871905 | Zbl 0632.60106

[20] J. Kigami. Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 (1995) 48-86. | MR 1317710 | Zbl 0820.60060

[21] J. Kigami. Analysis on Fractals. Cambridge Univ. Press, 2001. | MR 1840042 | Zbl 0998.28004

[22] W. B. Krebs. Brownian motion on the continuum tree. Probab. Theory Related Fields 101 (1995) 421-433. | MR 1324094 | Zbl 0822.60069

[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 2074701 | Zbl 1067.60070

[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 1188037 | Zbl 0762.60068

[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 655268 | Zbl 0456.60029

[26] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. | MR 1725357 | Zbl 0917.60006