A diffusion process in a Brownian environment with drift

Kiyoshi KAWAZU; Hiroshi TANAKA

doi:10.2969/jmsj/04920189

Kiyoshi KAWAZU, Hiroshi TANAKA

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JOURNAL FREE ACCESS

1997 Volume 49 Issue 2 Pages 189-211

DOI https://doi.org/10.2969/jmsj/04920189

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Published: 1997 Received: February 08, 1995 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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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) A. O. Golosov, Limit distributions for random walks in random environments, Soviet Math. Dokl., 28 (1983), 18-22.
3) A. O. Golosov, Localization of random walks in one-dimensional random environments, Comm. Math. Phys., 92 (1984), 491-506.
4) K. Ito, Stochastic Processes II, Iwanami, 1958, (in Japanese).
5) K. Ito and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4-1 (1964), 1-75.
6) I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. (2), 103 (1974), 19-102.
7) Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan J. Math., 1 (1975), 67-84.
8) K. Kawazu, Y. Tamura and H. Tanaka, Localization of diffusion processes in one-dimensional random environment, J. Math. Soc. Japan, 44 (1992), 515-550.
9) K. Kawazu and H. Tanaka, On the maximum of a diffusion process in a drifted Brownian environment, In Séminaire de Probabilités, (eds. J. Azéma et al.), Lecture Notes in Math., 1557, Springer-Verlag, 1993, 78-85.
10) H. Kesten, M. V. Kozlov and F. Spitzer, A limit law for random walk in a random environment, Compositio Math., 30 (1975), 145-168.
11) H. Kesten, The limit distribution of Sinai's random walk in random environment, Physica, 138A (1986), 299-309.
12) F. B. Knight, Characterization of the Lévy measure of inverse local times of gap diffusions, In Seminar on Stochastic Processes, Birkhäuser, 1981, 53-78.
13) S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, In Functional Analysis in Markov Processes, (ed. M. Fukushima), Lecture Notes in Math., 923, Springer-Verlag, 1982, 235-259.
14) M. V. Kozlov, A random walk on the line with stochastic structure, Theory Probab. Appl., 18 (1973), 406-408.
15) Y. Ogura, One-dimensional bi-generalized diffusion processes, J. Math. Soc. Japan, 41 (1989), 213-242.
16) S. Schumacher, Diffusions with random coefficients, Particle Systems, Random Media and Large Deviations, Contemp. Math., 41 (1985), 351-356.
17) Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl., 27 (1982), 256-268.
18) F. Solomon, Random walks in a random environment, Ann. Probab., 3 (1975), 1-31.
19) H. Tanaka, Localization of a diffusion process in a one-dimensional Brownian environment, Comm. Pure Appl. Math., XLVII (1994), 755-766.
20) S. Watanabe, On time inversion of one-dimensional diffusion processes, Z. Wahr-scheinlichtkeitstheorie verw. Gebiete, 31 (1975), 115-124.

Right : [1] T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14 (1986), 1206-1218.
[2] A. O. Golosov, Limit distributions for random walks in random environments, Soviet Math. Dokl., 28 (1983), 18-22.
[3] A. O. Golosov, Localization of random walks in one-dimensional random environments, Comm. Math. Phys., 92 (1984), 491-506.
[4] K. Itô, Stochastic Processes II, Iwanami, 1958, (in Japanese).
[5] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4-1 (1964), 1-75.
[6] I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. (2), 103 (1974), 19-102.
[7] Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan J. Math., 1 (1975), 67-84.
[8] K. Kawazu, Y. Tamura and H. Tanaka, Localization of diffusion processes in one-dimensional random environment, J. Math. Soc. Japan, 44 (1992), 515-550.
[9] K. Kawazu and H. Tanaka, On the maximum of a diffusion process in a drifted Brownian environment, In Séminaire de Probabilités, (eds. J. Azéma et al.), Lecture Notes in Math., 1557, Springer-Verlag, 1993, 78-85.
[10] H. Kesten, M. V. Kozlov and F. Spitzer, A limit law for random walk in a random environment, Compositio Math., 30 (1975), 145-168.
[11] H. Kesten, The limit distribution of Sinai's random walk in random environment, Physica, 138A (1986), 299-309.
[12] F. B. Knight, Characterization of the Lévy measure of inverse local times of gap diffusions, In Seminar on Stochastic Processes, Birkhäuser, 1981, 53-78.
[13] S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, In Functional Analysis in Markov Processes, (ed. M. Fukushima), Lecture Notes in Math., 923, Springer-Verlag, 1982, 235-259.
[14] M. V. Kozlov, A random walk on the line with stochastic structure, Theory Probab. Appl., 18 (1973), 406-408.
[15] Y. Ogura, One-dimensional bi-generalized diffusion processes, J. Math. Soc. Japan, 41 (1989), 213-242.
[16] S. Schumacher, Diffusions with random coefficients, Particle Systems, Random Media and Large Deviations, Contemp. Math., 41 (1985), 351-356.
[17] Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl., 27 (1982), 256-268.
[18] F. Solomon, Random walks in a random environment, Ann. Probab., 3 (1975), 1-31.
[19] H. Tanaka, Localization of a diffusion process in a one-dimensional Brownian environment, Comm. Pure Appl. Math., XLVII (1994), 755-766.
[20] S. Watanabe, On time inversion of one-dimensional diffusion processes, Z. Wahrscheinlichtkeitstheorie verw. Gebiete, 31 (1975), 115-124.

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