Asymptotic behavior of the first exit time of randomly perturbed dynamical systems with a repulsive equilibrium point

Toshio MIKAMI

doi:10.2969/jmsj/05010095

Toshio MIKAMI

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JOURNAL FREE ACCESS

1998 Volume 50 Issue 1 Pages 95-117

DOI https://doi.org/10.2969/jmsj/05010095

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Published: 1998 Received: April 27, 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) Day, M. V., On the exponential exit law in the small parameter exit problem. Stochastics. 8 297-323 (1983).
2) Day, M. V., Recent progress on the small parameter exit problem. Stochastics. 20 121-150 (1987).
3) Day, M. V., Cycling and skewing of exit measures for planar systems. Stochastics. 48 227-247 (1994).
4) Day, M. V., On the exit law from saddle points. preprint.
5) Fleming, W. H. and James, M. R., Asymptotic series and exit time probabilities. Ann. Probab. 20 1369-1384 (1992).
6) Fleming, W. H. and Soner, H. M., Controlled Markov processes and Viscosity solutions (Applications of Math. Bd. 25) Berlin Heidelberg New York Tokyo: Springer 1993.
7) Freidlin, M. I., Functional integration and partial differential equations. Princeton: Princeton University Press 1985.
8) Freidlin, M. I. and Wentzell, A. D., Random perturbations of dynamical systems (Grundlehren Math. Wiss. Bd. 260) Berlin Heidelberg New York Tokyo: Springer 1984.
9) Galves, A., Olivieri, E. and Vares, M. E., Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15 1288-1305 (1987).
10) Hartman, P., Ordinary differential equations. New York London Sydney: John Wiley & Sons 1964.
11) Ikeda, N. and Watanabe, S., Stochastic differential equations and diffusion processes, 2nd ed. Amsterdam New York Oxford Tokyo: North-Holland/Kodansha 1989.
12) Kifer, Y., The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Israel J. Math. 40 74-96 (1981).
13) Kifer, Y., Random perturbations of dynamical systems. Boston Basel: Birkhäuser 1988. Asymptotic behavior of the first exit time 117
14) Kifer, Y., Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states. Israel J. Math. 70 1-47 (1990).
15) Mikami, T., Limit theorems on the exit problems for small random perturbations of dynamical systems I. Stoc. and Stoc. Rep. 46 79-106 (1994).
16) Mikami, T., Large deviations for the first exit time on small random perturbations of dynamical systems with a hyperbolic equilibrium point. Hokkaido Math. J. 24 491-525 (1995).
17) Wentzell, A. D., Limit theorems on large deviations for Markov stochastic processes. Dordrecht Boston London: Kluwer Academic Publishers 1990.
18) Wentzell, A. D. and Freidlin, M. I., On small random perturbations of dynamical systems. Russian Math. Survey 25 1-55 (1970).
19) Wentzell, A. D. and Freidlin, M. I., Some problems concerning stability under small random perturbations. Theory Probab. Appl. 17 269-283 (1972).

Right : [1] Day, M. V., On the exponential exit law in the small parameter exit problem. Stochastics. 8 297-323 (1983).
[2] Day, M. V., Recent progress on the small parameter exit problem. Stochastics. 20 121-150 (1987).
[3] Day, M. V., Cycling and skewing of exit measures for planar systems. Stochastics. 48 227-247 (1994).
[4] Day, M. V., On the exit law from saddle points. preprint.
[5] Fleming, W. H. and James, M. R., Asymptotic series and exit time probabilities. Ann. Probab. 20 1369-1384 (1992).
[6] Fleming, W. H. and Soner, H. M., Controlled Markov processes and Viscosity solutions (Applications of Math. Bd. 25) Berlin Heidelberg New York Tokyo: Springer 1993.
[7] Freidlin, M. I., Functional integration and partial differential equations. Princeton: Princeton University Press 1985.
[8] Freidlin, M. I. and Wentzell, A. D., Random perturbations of dynamical systems (Grundlehren Math. Wiss. Bd. 260) Berlin Heidelberg New York Tokyo: Springer 1984.
[9] Galves, A., Olivieri, E. and Vares, M. E., Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15 1288-1305 (1987).
[10] Hartman, P., Ordinary differential equations. New York London Sydney: John Wiley & Sons 1964.
[11] Ikeda, N. and Watanabe, S., Stochastic differential equations and diffusion processes, 2nd ed. Amsterdam New York Oxford Tokyo: North-Holland/Kodansha 1989.
[12] Kifer, Y., The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Israel J. Math. 40 74-96 (1981).
[13] Kifer, Y., Random perturbations of dynamical systems. Boston Basel: Birkhäuser 1988.
[14] Kifer, Y., Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states. Israel J. Math. 70 1-47 (1990).
[15] Mikami, T., Limit theorems on the exit problems for small random perturbations of dynamical systems I. Stoc. and Stoc. Rep. 46 79-106 (1994).
[16] Mikami, T., Large deviations for the first exit time on small random perturbations of dynamical systems with a hyperbolic equilibrium point. Hokkaido Math. J. 24 491-525 (1995).
[17] Wentzell, A. D., Limit theorems on large deviations for Markov stochastic processes. Dordrecht Boston London: Kluwer Academic Publishers 1990.
[18] Wentzell, A. D. and Freidlin, M. I., On small random perturbations of dynamical systems. Russian Math. Survey 25 1-55 (1970).
[19] Wentzell, A. D. and Freidlin, M. I., Some problems concerning stability under small random perturbations. Theory Probab. Appl. 17 269-283 (1972).

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