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Limit theorems for one-dimensional diffusions and random walks in random environments
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  • Published: February 1989

Limit theorems for one-dimensional diffusions and random walks in random environments

  • Kiyoshi Kawazu1,
  • Yozo Tamura2 &
  • Hiroshi Tanaka2 

Probability Theory and Related Fields volume 80, pages 501–541 (1989)Cite this article

  • 196 Accesses

  • 18 Citations

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Summary

The limiting behavior of one-dimensional diffusion process in an asymptotically self-similar random environment is investigated through the extension of Brox's method. Similar problems are then discussed for a random walk in a random environment with the aid of optional sampling from a diffusion model; an extension of the result of Sinai is given in the case of asymptotically self-similar random environments.

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References

  1. Borsuk, K.: Theory of Retracts. Monografic Matematyczne, vol. 44, Polish Scientific Publishers Warsawa: 1967

    Google Scholar 

  2. Brox, T.: A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14, 1206–1218 (1986)

    Google Scholar 

  3. Golosov, A.O.: The limit distribution for random walks in random environments. Soviet. Math. Dokl. 28, 19–22 (1983)

    Google Scholar 

  4. Golosov, A.O.: Localization of random walks in one-dimensional random environments. Commun. Math. Phys. 92, 491–506 (1984)

    Google Scholar 

  5. Griffeath, D.: A maximal coupling for Markov chains. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 95–106 (1975)

    Google Scholar 

  6. Itô, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin Heidelberg: Springer 1965

    Google Scholar 

  7. Kawazu, K., Tamura, Y., Tanaka, H.: One-dimensional diffusions and random walks in random environments. In: Watanabe, S., Prokhorov, Yu. (eds.), Probability theory and mathematical statistics. (Lect. Notes Math., vol. 1299, pp. 170–184) Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  8. Kesten, H., Kozlov, M.V., Spitzer, F.: A limit law for random walk in a random environment. Compostio Math. 30, 145–168 (1975)

    Google Scholar 

  9. Kesten, H.: The limit distribution of Sinai's random walk in random environment. Physica 138A, 299–309 (1986)

    Google Scholar 

  10. Nadler, Jr., D.W.: Hyperspaces of sets. Monographs and textbooks in pure and applied mathematics, vol. 49. New York: Marcel Dekker 1978

    Google Scholar 

  11. Schumacher, S.: Diffusions with random coefficients. In: Durrett, R. (ed.) Contemporary mathematics: Particle systems, random media and large deviations, vol. 41, pp. 351–356 Providence: AMS 1985

    Google Scholar 

  12. Schumacher, S.: Diffusions with random coefficients. Ph.D. thesis. University of California, Los Angeles 1984

  13. Sinai, Y.G.: The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27, 256–268 (1982)

    Google Scholar 

  14. Tanaka, H.: Limit distributions for one-dimensional diffusion processes in self-similar random environments. In: Papanicolau, G. (ed.) IMA hydrodynamic behavior and interacting particle systems, vol. 9, pp. 189–210. Berlin Heidelberg New York: Springer 1987

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi, Japan

    Kiyoshi Kawazu

  2. Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan

    Yozo Tamura & Hiroshi Tanaka

Authors
  1. Kiyoshi Kawazu
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  2. Yozo Tamura
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  3. Hiroshi Tanaka
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Kawazu, K., Tamura, Y. & Tanaka, H. Limit theorems for one-dimensional diffusions and random walks in random environments. Probab. Th. Rel. Fields 80, 501–541 (1989). https://doi.org/10.1007/BF00318905

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  • Received: 25 May 1987

  • Revised: 21 June 1988

  • Issue Date: February 1989

  • DOI: https://doi.org/10.1007/BF00318905

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Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Diffusion Process
  • Limit Theorem
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