Transitions on a noncompact Cantor set and random walks on its defining tree

Kigami, Jun

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1090-1129.

Nous commençons par introduire des ensembles de Cantor non-compacts, ainsi que leurs arbres associés. Ils peuvent être considerés comme une généralisation naturelle des nombres $p$-adiques. Nous construisons ensuite une classe de processus de saut sur un ensemble de Cantor non-compact, à l’aide d’un couple de valeurs propres et de mesures. De plus, nous obtenons des expressions concrètes pour les noyaux de la chaleurs associés à ces processus de saut et pour les probabilités de transition correspondantes. Sous certaines hypothèses de régularité sur les valeurs propres et les mesures, nous construisons ensuite des métriques intrinsèques sur cet ensemble de Cantor non-compact afin d’obtenir des estimations fines sur les noyaux de la chaleur et les probabilités de transitions. Finalement, nous montrons que les marches aléatoires sur l’arbre définissant l’ensemble de Cantor non-compact induisent une sous-classe des processus de saut discutés dans la seconde partie de l’article.

First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of $p$-adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions on eigenvalues and measures. Finally transient random walks on the defining tree are shown to induce a subclass of jump processes discussed in the second part.

[1] S. Albeverio and W. Karwowski. A random walk on $p$-adics - the generator and its spectrum. Stochastic Process. Appl. 53 (1994) 1-22. | MR

[2] S. Albeverio and W. Karwowski. Jump processes on leaves of multibranching trees. J. Math. Phys. 49 (2008) 093503. | MR

[3] S. Albeverio, W. Karwowski and K. Yasuda. Trace formula for $p$-adics. Acta Appl. Math. 71 (2002) 31-48. | MR

[4] S. Albeverio, W. Karwowski and X. Zhao. Asymptotics and spectral results for random walks on $p$-adics. Stochastic Process. Appl. 83 (1999) 39-59. | MR

[5] D. Aldous and N. Evans. Dirichlet forms on totally disconnected spaces and bipartite Markov chains. J. Theor. Prob. 12 (1999) 839-857. | MR

[6] M. T. Barlow, A. Grigor'Yan and T. Kumagai. On the equivalence of parabolic harnack inequalities and heat kernel estimates. J. Math. Soc. Japan 64 (2012) 1091-1146. | MR

[7] M. T. Barlow, R. F. Bass and T. Kumagai. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan 58 (2006) 485-519. | MR

[8] R. M. Blumenthal and R. K. Getoor. Markov Processes and Potential Theory. Pure and Applied Mathematics 29. Academic Press, New York, 1968. | MR

[9] L. Brekke and P. G. O. Freund. $p$-adic numbers in physics. Phys. Rep. 233 (1993) 1-66. | MR

[10] P. Cartier. Fonctions harmoniques sur un arbre. In Sympos. Math., vol. 9 203-270. Academic Press, London, 1972. | MR

[11] Z. Q. Chen and T. Kumagai. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 (2008) 277-317. | MR

[12] B. Dragovich, A. Y. Khrennikov, S. V. Kozrev and I. V. Volovich. On $p$-adic mathematical physics. $p$-Adic Numbers, Ultrametric Anal. Appl. 1 (2009) 1-17. | MR

[13] N. Evans. Local properties of Lévy processes on a totally disconnected group. J. Theoret. Probab. 2 (1989) 209-259. | MR

[14] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Math. 19. de Gruyter, Berlin, 1994. | MR

[15] A. Grigor'Yan. The heat equation on noncompact Riemannian manifolds. (in Russian). Mat. Sb. 182 (1991) 55-87. English translation in Math. USSR-Sb. 72 (1992) 47-77. | MR

[16] A. Grigor'Yan and A. Telcs. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324 (2002) 521-556. | MR

[17] W. Karwowski and R. Vilea-Mendes. Hierarchical structures and assymetric process on $p$-adics and adeles. J. Math. Phys. 35 (1994) 4637-4650. | MR

[18] J. Kigami. Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees. Adv. Math. 225 (2010) 2674-2730. | MR

[19] R. Rammal, G. Toulouse and M. A. Virasoro. Ultametricity for physicists. Rev. Mod. Phys. 58 (1986) 765-788. | MR

[20] L. Saloff-Coste. A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices (1992) 27-38. | MR

[21] W. Woess. Random walks on infinite graphs and groups - a surveey on selected topics. Bull. London Math. Soc. 26 (1994) 1-60. | MR

[22] W. Woess. Denumerable Markov Chains. European Math. Soc., Zürich, 2009. | MR

[23] M. Yamasaki. Parabolic and hyperbolic infinite networks. Hiroshima Math. J. 7 (1977) 135-146. | MR

[24] M. Yamasaki. Discrete potentials on an infinite network. Mem. Fac. Sci. Shimane Univ. 13 (1979) 31-44. | MR