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Some Results and Problems for Anisotropic Random Walks on the Plane

  • Endre CsákiEmail author
  • Antónia Földes
  • Pál Révész
Part of the Fields Institute Communications book series (FIC, volume 76)

Abstract

This is an expository paper on the asymptotic results concerning path behaviour of the anisotropic random walk on the two-dimensional square lattice \(\mathbb{Z}^{2}\). In recent years Miklós and the authors of the present paper investigated the properties of this random walk concerning strong approximations, local times and range. We give a survey of these results together with some further problems.

Notes

Acknowledgements

The authors thank the referees for valuable comments and suggestions. Research supported by PSC CUNY Grant, No. 68080-0043 and by the Hungarian National Foundation for Scientific Research, No. K108615.

References

  1. 1.
    Bertacchi, D.: Asymptotic behaviour of the simple random walk on the 2-dimensional comb. Electron. J. Probab. 11, 1184–1203 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertacchi, D., Zucca, F.: Uniform asymptotic estimates of transition probabilities on combs. J. Aust. Math. Soc. 75, 325–353 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertoin, J.: Iterated Brownian Motion and stable (1/4) subordinator. Stat. Probab. Lett. 27, 111–114 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, X.: How often does a Harris recurrent Markov chain recur? Ann. Probab. 27, 1324–1346 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Csáki, E., Csörgő, M., Földes, A., Révész, P.: Strong limit theorems for a simple random walk on the 2-dimensional comb. Electron. J. Probab. 14, 2371–2390 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Csáki, E., Csörgő, M., Földes, A., Révész, P.: On the supremum of iterated local time. Publ. Math. Debr. 76, 255–270 (2010)zbMATHGoogle Scholar
  7. 7.
    Csáki, E., Csörgő, M., Földes, A., Révész, P.: On the local time of random walk on the 2-dimensional comb. Stoch. Process. Appl. 121, 1290–1314 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Csáki, E., Csörgő, M., Földes, A., Révész, P.: Random walk on half-plane half-comb structure. Annales Mathematicae et Informaticae. 39, 29–44 (2012)zbMATHGoogle Scholar
  9. 9.
    Csáki, E., Csörgő, M., Földes, A., Révész, P.: Strong limit theorems for anisotropic random walks on \(\mathbb{Z}^{2}\). Periodica Math. Hungar. 67, 71–94 (2013)CrossRefGoogle Scholar
  10. 10.
    Csáki, E., Földes, A., Révész, P.: Strassen theorems for a class of iterated processes. Trans. Am. Math. Soc. 349, 1153–1167 (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Darling, D.A., Kac, M.: On occupation times for Markoff processes. Trans. Am. Math. Soc. 84, 444–458 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    den Hollander, F.: On three conjectures by K. Shuler. J. Stat. Phys. 75, 891–918 (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dvoretzky, A., Erdős, P.: Some problems on random walk in space. In: Proceedings of Second Berkeley Symposium, Berkeley, pp. 353–367 (1951)Google Scholar
  14. 14.
    Heyde, C.C.: On the asymptotic behaviour of random walks on an anisotropic lattice. J. Stat. Phys. 27, 721–730 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Heyde, C.C.: Asymptotics for two-dimensional anisotropic random walks. In: Stochastic Processes, pp. 125–130. Springer, New York (1993)Google Scholar
  16. 16.
    Heyde, C.C., Westcott, M., Williams, E.R.: The asymptotic behavior of a random walk on a dual-medium lattice. J. Stat. Phys. 28, 375–380 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Horváth, L.: Diffusion approximation for random walks on anisotropic lattices. J. Appl. Probab. 35, 206–212 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Krishnapur, M., Peres, Y.: Recurrent graphs where two independent random walks collide finitely often. Electron. Commun. Probab. 9, 72–81 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lévy, P.: Processus Stochastiques et Mouvement Brownien. Gauthier Villars, Paris (1948)zbMATHGoogle Scholar
  20. 20.
    Nándori, P.: Number of distinct sites visited by a random walk with internal states. Probab. Theory Relat. Fields 150, 373–403 (2011)CrossRefzbMATHGoogle Scholar
  21. 21.
    Nash-Williams, C. St. J.A.: Random walks and electric currents in networks. Proc. Camb. Philos. Soc. 55, 181–194 (1959)Google Scholar
  22. 22.
    Petrov, V.V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  23. 23.
    Révész, P.: Random Walk in Random and Non-random Environments, 2nd edn. World Scientific, Singapore (2005)CrossRefzbMATHGoogle Scholar
  24. 24.
    Revuz, D.: Markov Chain. North-Holland, Amsterdam (1975)Google Scholar
  25. 25.
    Roerdink, J., Shuler, K.E.: Asymptotic properties of multistate random walks. I. Theory. J. Stat. Phys. 40, 205–240 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Seshadri, V., Lindenberg, K., Shuler, K.E.: Random walks on periodic and random lattices. II. Random walk properties via generating function techniques. J. Stat. Phys. 21, 517–548 (1979)Google Scholar
  27. 27.
    Shuler, K.E.: Random walks on sparsely periodic and random lattices I. Physica A 95, 12–34 (1979)CrossRefGoogle Scholar
  28. 28.
    Silver, H., Shuler, K.E., Lindenberg, K.: Two-dimensional anisotropic random walks. In: Statistical Mechanics and Statistical Methods in Theory and Application (Proc. Sympos., Univ. Rochester, Rochester, 1976). Plenum, New York, pp. 463–505 (1977)Google Scholar
  29. 29.
    Weiss, G.H., Havlin, S.: Some properties of a random walk on a comb structure. Physica A 134, 474–482 (1986)CrossRefGoogle Scholar
  30. 30.
    Westcott, M.: Random walks on a lattice. J. Stat. Phys. 27, 75–82 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestsHungary
  2. 2.Department of MathematicsCollege of Staten Island, CUNYNew YorkUSA
  3. 3.Institut für Statistik und WahrscheinlichkeitstheorieTechnische Universität WienViennaAustria

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