Recent progress on the small parameter exit problemFootnote^†

^†This work was supported in part by the National Science Foundation under grant DMS-8420755.

References

• Day , M. V. 1986 . The exterior sphere condition for the Ventcel-Freidlin quasi-potential function . Appl. Math. Opt , 14 : 49 – 54 .
   Web of Science ®Google Scholar
• Day , M. V. 1984 . On the asymptotic relation between equilibrium density and exit measure in the exit problem . Stochastics , 12 : 303 – 330 .
   Google Scholar
• Day , M. V. and Darden , T.A . 1985 . Some regularity results on the Ventcel-Freidlin quasi-potential function . Appl. Math. Opt , 13 : 259 – 282 .
   Web of Science ®Google Scholar
• Devinatz , A. and Friedman , A . 1978 . The asymptotic behavior of the solution of a singularly perturbed Dirichlet problem . Indiana J. Math , 2 : 527 – 537 .
   Google Scholar
• Fleming , W. H. 1978 . Exit probabilities and optimal stochastic control . Appl. Math.Opt , 4 : 329 – 346 .
   Web of Science ®Google Scholar
• Freidlin , M. I. and Wentzell , A.D . 1984 . Random Perturbations of Dynamical Systems , New York : Springer .
   Google Scholar
• Friedman , A. 1976 . Stochastic Differential Equations and Applications , New York : AcademicPress .
   Google Scholar
• Hasminskii , R. Z. 1980 . Stochastic Stability of Differential Equations , Alphen aanden Rijn, , Netherlands : Sijthoff and Noordoff Intl .
   Google Scholar
• Hwang , S. 1980 . Laplace's method revisited:weak convergence of probability measures . Ann. of Prob , 8 : 1177 – 1182 .
   Web of Science ®Google Scholar
• Kamin , S. 1978 . Elliptic perturbation of a first order operator with a singular point of attracting type . Indiana U. Math.J , 27 : 935 – 952 .
   Web of Science ®Google Scholar
• Kifer , Y. I. 1974 . Certain results concerning small random perturbations of dynamical systems . Th. Prob. and Appl , 19 : 487 – 505 .
   Google Scholar
• Ludwig , D. 1975 . Persistence of dynamical systems under random perturbations . SIAM Review , 17 : 605 – 640 .
   Web of Science ®Google Scholar
• Matkowsky , B. J. and Schuss , Z . 1979 . The exit problem for randomly perturbed dynamical systems . SIAMJ. Appl. Math , 36 : 365 – 382 .
   Google Scholar
• Matkowsky , B. J. , Schuss , Z and Tier , C . 1983 . Diffusion across characteristic boundaries with critical points . SIAM J. Appl. Math , 43 : 673 – 695 .
   Web of Science ®Google Scholar
• Schuss , Z. 1980 . Theory and Applications of Stochastic Differential Equations , New York : J. Wileyand Sons .
   Google Scholar
• Schuss , Z. and Matkowsky , B.J . 1979 . The exit problem:a new approach to diffusion across potential barriers . SIAM J. Appl. Math , 35 : 604 – 623 .
   Google Scholar
• Sheu , S. J. 1986 . Asymptotic behavior of invariant density of diffusion Markov process with small diffusion . SIAM J. Math. Anal , 17 : 451 – 460 .
   Web of Science ®Google Scholar
• Varadhan , S. R. S . 1984 . Large Deviations and Applications . CBMS-NSF Regional Conference Series on Applied Mathematics . 1984 , Philadelphia. Vol. 46 , SIAM
   Google Scholar
• Ventcel , A. D and Freidlin , M.I . 1970 . On small random perturbations of dynamical systems . Uspei. Mat. Nauk , 25 : 1 – 56 . Russian Math. Surveys 25 ‘1970’,1-55
   Google Scholar