Localization of diffusion processes in one-dimensional random environment

Kiyoshi KAWAZU; Yozo TAMURA; Hiroshi TANAKA

doi:10.2969/jmsj/04430515

Kiyoshi KAWAZU, Yozo TAMURA, Hiroshi TANAKA

Author information

JOURNAL FREE ACCESS

1992 Volume 44 Issue 3 Pages 515-550

DOI https://doi.org/10.2969/jmsj/04430515

Details

Published: 1992 Received: July 18, 1990 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
Correction information
Date of correction: October 20, 2006 Reason for correction: - Correction: AFFILIATION Details: Wrong :
1) Department of MathematicsFaculty of Education Yamaguchi University
2) Department of Mathematics Faculty of Science and Technology Keio University

Right :
1) Department of Mathematics Faculty of Education Yamaguchi University
2) Department of Mathematics Faculty of Science and Technology Keio University

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14 (1986), 1206-1218.
2) K. L. Chung, A Course in Probability Theory, 2nd ed., Academic Press, New York, 1974.
3) W. Feller, An Introduction to Probability Theory and Its Applications Vol. II, 2nd ed., John Wiley, New York, 1971.
4) A. O. Golosov, Localization of random walks in one-dimensional random environments, Commun. Math. Phys., 92 (1984), 491-506.
5) A. O. Golosov, On limiting distributions for a random walk in a critical one-dimensional random environment, Russian Math. Survey, 41 (1986), 199-200.
6) K. Ito and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965.
7) K. Kawazu, Y. Tamura and H. Tanaka, One-dimensional diffusions and random walks in random environments, Probab. Th. Math. Statist. 5th Japan-U. S. S. R. Symposium Proceedings, 1986, (eds. S. Watanabe and Yu. V. Prokhorov), Lecture Notes in Math., 1299, Springer, 1988, pp. 170-184.
8) K. Kawazu, Y. Tamura and H. Tanaka, Limit theorems for one-dimensional diffusions and random walks in random environments, Probab. Th. Rel. Fields, 80 (1989), 501-541.
9) H. Kesten, The limit distribution of Sinai's random walk in random environment, Physica, 138A (1986), 299-309.
10) A. V. Letchikov, Localization of one-dimensional random walk in a random environments, Soviet Scientific Reviews sect. C, Mathematical Physics Reviews, Harwood Academic Publishers, New York, 1989.
11) B. A. Rogozin, The distribution of the first ladder moment and height and fluctuation of a random walk, Theory Probab. Appl., 16 (1971), 575-595.
12) Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl., 27 (1982), 256-268.
13) A. V. Skorohod, Limit theorems for stochastic processes with independent increments, Theory Probab. Appl., 2 (1957), 138-171.
14) F. Spitzer, A Tauberian theorem and its probability interpretation, Trans. Amer. Math. Soc., 94 (1960), 150-169.
15) H. Tanaka, Limit distributions for one-dimensional diffusion processes in self-similar random environments, IMA hydrodynamic behavior and interacting particle systems, vol. 9, (ed. G. Papanicolaou), Springer, New York, 1987, pp. 189-210.
16) H. Tanaka, Limit theorem for one-dimensional diffusion process in Brownian environment, Stochastic Analysis, Proc. Japanese-French Seminar held in Paris, 1987, Lecture Notes in Math, 1322, Springer, pp. 156-172.
17) H. Tanaka, Time reversal of random walks in one-dimension, Tokyo J. Math., 12 (1989), 159-174.

Right : [1] T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14 (1986), 1206-1218.
[2] K. L. Chung, A Course in Probability Theory, 2nd ed., Academic Press, New York, 1974.
[3] W. Feller, An Introduction to Probability Theory and Its Applications Vol. II, 2nd ed., John Wiley, New York, 1971.
[4] A. O. Golosov, Localization of random walks in one-dimensional random environments, Commun. Math. Phys., 92 (1984), 491-506.
[5] A. O. Golosov, On limiting distributions for a random walk in a critical one-dimensional random environment, Russian Math. Survey, 41 (1986), 199-200.
[6] K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965.
[7] K. Kawazu, Y. Tamura and H. Tanaka, One-dimensional diffusions and random walks in random environments, Probab. Th. Math. Statist. 5th Japan-U. S. S. R. Symposium Proceedings, 1986, (eds. S. Watanabe and Yu. V. Prokhorov), Lecture Notes in Math., 1299, Springer, 1988, pp. 170-184.
[8] K. Kawazu, Y. Tamura and H. Tanaka, Limit theorems for one-dimensional diffusions and random walks in random environments, Probab. Th. Rel. Fields, 80 (1989), 501-541.
[9] H. Kesten, The limit distribution of Sinai's random walk in random environment, Physica, 138A (1986), 299-309.
[10] A. V. Letchikov, Localization of one-dimensional random walk in a random environments, Soviet Scientific Reviews sect. C, Mathematical Physics Reviews, Harwood Academic Publishers, New York, 1989.
[11] B. A. Rogozin, The distribution of the first ladder moment and height and fluctuation of a random walk, Theory Probab. Appl., 16 (1971), 575-595.
[12] Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl., 27 (1982), 256-268.
[13] A. V. Skorohod, Limit theorems for stochastic processes with independent increments, Theory Probab. Appl., 2 (1957), 138-171.
[14] F. Spitzer, A Tauberian theorem and its probability interpretation, Trans. Amer. Math. Soc., 94 (1960), 150-169.
[15] H. Tanaka, Limit distributions for one-dimensional diffusion processes in self-similar random environments, IMA hydrodynamic behavior and interacting particle systems, vol. 9, (ed. G. Papanicolaou), Springer, New York, 1987, pp. 189-210.
[16] H. Tanaka, Limit theorem for one-dimensional diffusion process in Brownian environment, Stochastic Analysis, Proc. Japanese-French Seminar held in Paris, 1987, Lecture Notes in Math, 1322, Springer, pp. 156-172.
[17] H. Tanaka, Time reversal of random walks in one-dimension, Tokyo J. Math., 12 (1989), 159-174.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Corresponding author

Correction information

Register with J-STAGE for free!