Metastable behaviors of diffusion processes with small parameter

Makoto SUGIURA

doi:10.2969/jmsj/04740755

Makoto SUGIURA

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JOURNAL FREE ACCESS

1995 Volume 47 Issue 4 Pages 755-788

DOI https://doi.org/10.2969/jmsj/04740755

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Published: 1995 Received: July 14, 1993 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) M. V. Day, On the exponential exit law in the small parameter exit problem, Stochastics, 8 (1983), 297-323.
2) A. Devinatz and A. Friedman, Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem, Indiana Univ. Math. J., 27 (1978), 143-157.
3) S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, John Wiley & Sons, New York, 1986.
4) M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, Berlin-Heidelberg-New York, 1984.
5) A. Friedman, Stochastic Differential Equations and Applications, Volume 2, Academic Press, New York, 1976.
6) A. Galves, E. Olivieri and M. E. Vares, Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab., 15 (1987), 1288-1305.
7) D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1983.
8) A. Gray, Tubes, Addison-Wesley, Redwood City, 1990.
9) S. Kamin, On elliptic singular perturbation problems with turning points, SIAM J. Math. Anal., 10 (1979), 447-455.
10) C. Kipnis and C. M. Newman, The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes, SIAM J. Appl. Math., 45 (1985), 972-982.
11) F. Martinelli, E. Olivieri and E. Scoppola, Small random perturbations of finite- and infinite-dimensional dynamical systems: unpredictability of exit times, J. Statist. Phys., 55 (1989), 477-504.
12) B. J. Matkowsky and Z. Schuss, The exit problem for randomly perturbed dynamical systems, SIAM J. Appl. Math., 33 (1977), 365-382.
13) Y. Ogura, One-dimensional bi-generalized diffusion processes, J. Math. Soc. Japan, 41 (1989), 213-242.
14) M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volumes II, IV, Academic Press, New York, 1975, 1978.
15) Z. Schuss, Theory and Applications of Stochastic Differential Equations, John Wiley & Sons, New York, 1980.
16) M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, 2nd ed., Publish or Perish, Wilmington, 1979.
17) M. Sugiura, Exponential asymptotics in the small parameter exit problem, preprint, 1993, to appear in Nagoya Math. J..
18) M. Sugiura, Limit theorems related to the small parameter exit problems and the singularly perturbed Dirichlet problems, preprint, 1994.
19) M. Williams, Asymptotic exit time distributions, SIAM J. Appl. Math., 42 (1982), 149-154.

Right : [1] M. V. Day, On the exponential exit law in the small parameter exit problem, Stochastics, 8 (1983), 297-323.
[2] A. Devinatz and A. Friedman, Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem, Indiana Univ. Math. J., 27 (1978), 143-157.
[3] S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, John Wiley & Sons, New York, 1986.
[4] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, Berlin-Heidelberg-New York, 1984.
[5] A. Friedman, Stochastic Differential Equations and Applications, Volume 2, Academic Press, New York, 1976.
[6] A. Galves, E. Olivieri and M. E. Vares, Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab., 15 (1987), 1288-1305.
[7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1983.
[8] A. Gray, Tubes, Addison-Wesley, Redwood City, 1990.
[9] S. Kamin, On elliptic singular perturbation problems with turning points, SIAM J. Math. Anal., 10 (1979), 447-455.
[10] C. Kipnis and C. M. Newman, The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes, SIAM J. Appl. Math., 45 (1985), 972-982.
[11] F. Martinelli, E. Olivieri and E. Scoppola, Small random perturbations of finite- and infinite-dimensional dynamical systems: unpredictability of exit times, J. Statist. Phys., 55 (1989), 477-504.
[12] B. J. Matkowsky and Z. Schuss, The exit problem for randomly perturbed dynamical systems, SIAM J. Appl. Math., 33 (1977), 365-382.
[13] Y. Ogura, One-dimensional bi-generalized diffusion processes, J. Math. Soc. Japan, 41 (1989), 213-242.
[14] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volumes II, IV, Academic Press, New York, 1975, 1978.
[15] Z. Schuss, Theory and Applications of Stochastic Differential Equations, John Wiley & Sons, New York, 1980.
[16] M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, 2nd ed., Publish or Perish, Wilmington, 1979.
[17] M. Sugiura, Exponential asymptotics in the small parameter exit problem, preprint, 1993, to appear in Nagoya Math. J..
[18] M. Sugiura, Limit theorems related to the small parameter exit problems and the singularly perturbed Dirichlet problems, preprint, 1994.
[19] M. Williams, Asymptotic exit time distributions, SIAM J. Appl. Math., 42 (1982), 149-154.

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

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