One-dimensional bi-generalized diffusion processes

Yukio OGURA

doi:10.2969/jmsj/04120213

Yukio OGURA

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JOURNAL FREE ACCESS

1989 Volume 41 Issue 2 Pages 213-242

DOI https://doi.org/10.2969/jmsj/04120213

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Published: 1989 Received: September 14, 1987 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: SUBTITLE Details: Wrong : Dedicated to Professor Nobuyuki Ikeda on his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) S. Albeverio, S. Kusuoka and L. Streit, Convergence of Dirichlet forms and associated Schrödinger operators, J. Funct. Analysis, 68 (1986), 130-148.
2) T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14 (1986), 1206-1218.
3) C. Dellacherie and P. A. Meyer, Probabilities and Potential B, North-Holland, Amsterdam, New York, Oxford, 1980.
4) H. Dym and H. P. McKean, Gaussian Processes, Function Theory and the Inverse Spectral Problem, Academic Press, New York, San Francisco, London, 1976.
5) W. Feller, The birth and death processes as diffusion processes, J. Math. Pures Appl., tome XXXVIII (1959), 301-345.
6) J. H. Gillespie, A simple stochastic gene substitution model, Theoret. Population Biol., 23 (1983), 202-215.
7) A. O. Golosov, Limit theorems for random walks in symmetric random environments, Theory Probab. Appl., XXIX (1984), 266-280.
8) M. Iizuka and Y. Ogura, Convergence of one-dimensional diffusion processes to a jump process related to population genetics, in preparation.
9) K. Ito and H. P. McKean, Jr., Diffusion Processes and their Sample Paths, Springer, 1965.
10) I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. Ser. 2, 103 (1974), 19-102.
11) S. Karlin and J. L. McGregor, The differential equations and birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc., 85 (1957), 489-546.
12) K. Kawazu and H. Kesten, On birth and death processes in symmetric random environment, J. Statist. Phys., 37 (1983), 305-314.
13) K. Kawazu, Y. Ogura and Y. Takahashi, A remark on one-dimensional diffusion processes with discontinuous scale functions, 15-th Conference on Stochastic Processes and their Applications, Nagoya, 1985: Abstracts in Stochastic Processes Appl., 21 (1985), 63.
14) K. Kawazu, Y. Tamura and H. Tanaka, One-dimensional diffusions and random walks in random environments, Probability Theory and Mathematical Statistics, ed. by S. Watanabe and Yu. V. Prohorov, Lecture Notes in Math., 1299, Springer, 1988, pp. 170-184.
15) K. Kipnis and C. M. Newman, The metastable behavior of infrequently observed random, one-dimensional diffusion processes, SIAM J. Appl. Math., 45 (1985), 972-982.
16) F. Knight, Note on regularlization of Markov processes, Illinois J. Math., 9 (1935), 548-552.
17) S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes. Functional Analysis in Markov Processes, ed. by M. Fukushima, Lecture Notes in Math., 923, Springer, 1982, pp. 235-259.
18) H. P. McKean, Jr., Elementary solutions for certain parabolic differential equations, Trans. Amer. Math. Soc., 82 (1956), 519-548.
19) N. Minami, Y. Ogura and M. Tomisaki, Asymptotic behavior of elementary solutions of one-dimensional generalized diffusion equations, Ann. Probability, 13 (1985), 698-715.
20) D. B. Ray, Resolvents, transition functions and strong Markovian processes, Ann. Math., 70 (1959), 43-72.
21) H. Tanaka, Limit distribution for one-dimensional diffusion processes in self-similar random environments, IMA Volume 9 Hydrodynamic Behavior and Interacting Particle Systems, Springer, 1987, pp. 189-210.
22) H. Tanaka, Limit distribution for 1-dimensional diffusion in a reflected Brownian medium, Séminaire de Probabilités XXI, ed. by J. Azema, P. A. Meyer and M. Yor, Lecture Notes in Math., 1248, Springer, 1987, pp. 246-261.
23) A. D. Ventsel and M. L. Freidlin, On small random perturbation of dynamic systems, Russ. Math. Surv., 25 (1970), 1-55.
24) D. Williams, Diffusions, Markov Processes, & Martingales. Vol. 1: Foundations, John Wiley, Chichester, New York, Brisbane, Toronto, 1979.

Right : [1] S. Albeverio, S. Kusuoka and L. Streit, Convergence of Dirichlet forms and associated Schrödinger operators, J. Funct. Analysis, 68 (1986), 130-148.
[2] T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14 (1986), 1206-1218.
[3] C. Dellacherie and P. A. Meyer, Probabilities and Potential B, North-Holland, Amsterdam, New York, Oxford, 1980.
[4] H. Dym and H. P. McKean, Gaussian Processes, Function Theory and the Inverse Spectral Problem, Academic Press, New York, San Francisco, London, 1976.
[5] W. Feller, The birth and death processes as diffusion processes, J. Math. Pures Appl., tome XXXVIII (1959), 301-345.
[6] J. H. Gillespie, A simple stochastic gene substitution model, Theoret. Population Biol., 23 (1983), 202-215.
[7] A. O. Golosov, Limit theorems for random walks in symmetric random environments, Theory Probab. Appl., XXIX (1984), 266-280.
[8] M. Iizuka and Y. Ogura, Convergence of one-dimensional diffusion processes to a jump process related to population genetics, in preparation.
[9] K. Itô and H. P. McKean, Jr., Diffusion Processes and their Sample Paths, Springer, 1965.
[10] I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. Ser. 2, 103 (1974), 19-102.
[11] S. Karlin and J. L. McGregor, The differential equations and birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc., 85 (1957), 489-546.
[12] K. Kawazu and H. Kesten, On birth and death processes in symmetric random environment, J. Statist. Phys., 37 (1983), 305-314.
[13] K. Kawazu, Y. Ogura and Y. Takahashi, A remark on one-dimensional diffusion processes with discontinuous scale functions, 15-th Conference on Stochastic Processes and their Applications, Nagoya, 1985: Abstracts in Stochastic Processes Appl., 21 (1985), 63.
[14] K. Kawazu, Y. Tamura and H. Tanaka, One-dimensional diffusions and random walks in random environments, Probability Theory and Mathematical Statistics, ed. by S. Watanabe and Yu. V. Prohorov, Lecture Notes in Math., 1299, Springer, 1988, pp. 170-184.
[15] K. Kipnis and C. M. Newman, The metastable behavior of infrequently observed random, one-dimensional diffusion processes, SIAM J. Appl. Math., 45 (1985), 972-982.
[16] F. Knight, Note on regularlization of Markov processes, Illinois J. Math., 9 (1935), 548-552.
[17] S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes. Functional Analysis in Markov Processes, ed. by M. Fukushima, Lecture Notes in Math., 923, Springer, 1982, pp. 235-259.
[18] H. P. McKean, Jr., Elementary solutions for certain parabolic differential equations, Trans. Amer. Math. Soc., 82 (1956), 519-548.
[19] N. Minami, Y. Ogura and M. Tomisaki, Asymptotic behavior of elementary solutions of one-dimensional generalized diffusion equations, Ann. Probability, 13 (1985), 698-715.
[20] D. B. Ray, Resolvents, transition functions and strong Markovian processes, Ann. Math., 70 (1959), 43-72.
[21] H. Tanaka, Limit distribution for one-dimensional diffusion processes in self-similar random environments, IMA Volume 9 Hydrodynamic Behavior and Interacting Particle Systems, Springer, 1987, pp. 189-210.
[22] H. Tanaka, Limit distribution for 1-dimensional diffusion in a reflected Brownian medium, Séminaire de Probabilités XXI, ed. by J. Azema, P. A. Meyer and M. Yor, Lecture Notes in Math., 1248, Springer, 1987, pp. 246-261.
[23] A. D. Ventsel and M. L. Freidlin, On small random perturbation of dynamic systems, Russ. Math. Surv., 25 (1970), 1-55.
[24] D. Williams, Diffusions, Markov Processes, & Martingales. Vol. 1: Foundations, John Wiley, Chichester, New York, Brisbane, Toronto, 1979.

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

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