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

Jun Kigami

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 1090-1129
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Kigami, Jun. "Transitions on a noncompact Cantor set and random walks on its defining tree." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 1090-1129. <http://eudml.org/doc/271965>.

@article{Kigami2013,
abstract = {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.},
author = {Kigami, Jun},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {noncompact Cantor set; $p$-adic numbers; tree; jump process; Dirichlet forms; random walks; Martin boundary; Dirichlet forms; noncompact Cantor set; -adic numbers; tree; jump process; random walks; Martin boundary},
language = {eng},
number = {4},
pages = {1090-1129},
publisher = {Gauthier-Villars},
title = {Transitions on a noncompact Cantor set and random walks on its defining tree},
url = {http://eudml.org/doc/271965},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Kigami, Jun
TI - Transitions on a noncompact Cantor set and random walks on its defining tree
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1090
EP - 1129
AB - 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.
LA - eng
KW - noncompact Cantor set; $p$-adic numbers; tree; jump process; Dirichlet forms; random walks; Martin boundary; Dirichlet forms; noncompact Cantor set; -adic numbers; tree; jump process; random walks; Martin boundary
UR - http://eudml.org/doc/271965
ER -

References

top
  1. [1] S. Albeverio and W. Karwowski. A random walk on p -adics – the generator and its spectrum. Stochastic Process. Appl.53 (1994) 1–22. Zbl0810.60065MR1290704
  2. [2] S. Albeverio and W. Karwowski. Jump processes on leaves of multibranching trees. J. Math. Phys. 49 (2008) 093503. Zbl1152.81310MR2455842
  3. [3] S. Albeverio, W. Karwowski and K. Yasuda. Trace formula for p -adics. Acta Appl. Math.71 (2002) 31–48. Zbl1044.11045MR1893360
  4. [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. Zbl0999.60072MR1705599
  5. [5] D. Aldous and N. Evans. Dirichlet forms on totally disconnected spaces and bipartite Markov chains. J. Theor. Prob.12 (1999) 839–857. Zbl0945.60064MR1702871
  6. [6] M. T. Barlow, A. Grigor’yan and T. Kumagai. On the equivalence of parabolic harnack inequalities and heat kernel estimates. J. Math. Soc. Japan64 (2012) 1091–1146. Zbl1281.58016MR2998918
  7. [7] M. T. Barlow, R. F. Bass and T. Kumagai. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan58 (2006) 485–519. Zbl1102.60064MR2228569
  8. [8] R. M. Blumenthal and R. K. Getoor. Markov Processes and Potential Theory. Pure and Applied Mathematics 29. Academic Press, New York, 1968. Zbl0169.49204MR264757
  9. [9] L. Brekke and P. G. O. Freund. p -adic numbers in physics. Phys. Rep.233 (1993) 1–66. MR1238475
  10. [10] P. Cartier. Fonctions harmoniques sur un arbre. In Sympos. Math., vol. 9 203–270. Academic Press, London, 1972. Zbl0283.31005MR353467
  11. [11] Z. Q. Chen and T. Kumagai. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields140 (2008) 277–317. Zbl1131.60076MR2357678
  12. [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. Zbl1187.81004MR2566116
  13. [13] N. Evans. Local properties of Lévy processes on a totally disconnected group. J. Theoret. Probab.2 (1989) 209–259. Zbl0683.60010MR987578
  14. [14] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Math. 19. de Gruyter, Berlin, 1994. MR1303354
  15. [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. MR1098839
  16. [16] A. Grigor’yan and A. Telcs. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann.324 (2002) 521–556. Zbl1011.60021MR1938457
  17. [17] W. Karwowski and R. Vilea-Mendes. Hierarchical structures and assymetric process on p -adics and adeles. J. Math. Phys.35 (1994) 4637–4650. Zbl0814.60103MR1290892
  18. [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. Zbl1234.60077MR2680180
  19. [19] R. Rammal, G. Toulouse and M. A. Virasoro. Ultametricity for physicists. Rev. Mod. Phys.58 (1986) 765–788. MR854445
  20. [20] L. Saloff-Coste. A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices (1992) 27–38. Zbl0769.58054MR1150597
  21. [21] W. Woess. Random walks on infinite graphs and groups – a surveey on selected topics. Bull. London Math. Soc.26 (1994) 1–60. Zbl0830.60061MR1246471
  22. [22] W. Woess. Denumerable Markov Chains. European Math. Soc., Zürich, 2009. MR2548569
  23. [23] M. Yamasaki. Parabolic and hyperbolic infinite networks. Hiroshima Math. J.7 (1977) 135–146. Zbl0382.90088MR429377
  24. [24] M. Yamasaki. Discrete potentials on an infinite network. Mem. Fac. Sci. Shimane Univ.13 (1979) 31–44. Zbl0416.31012MR558311

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.