Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential

Annals of Probability

Volumn 25, Issue 4, 1997, Pages 1953-1994

  Fleming, Wendell H ^a   Sheu, Shuenn Jyi ^a  

^a NONE

Author keywords

Diffusion processes with small noise; Discounted control problem; First eigenvalue and eigenfunction; Large deviations; Risk sensitive control; Viscosity solution

Indexed keywords

EID: 0031320509     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1023481117     Document Type: Article

References (46)

1. n. J. Reine Angew. Math. 429 125-160.
2. BENSOUSSAN, A. and NAGAI, H. (1991). An ergodic control problem arising from the principal eigenfunction of an elliptic operator. Publ. Math. Soc. Japan 43 49-65.
3. DAY, M. V. (1987). Recent progress on the small parameter exit problem. Stochastics 20 121-150.
4. DAY, M. V. and DARDEN, T. A. (1985). Some regularity results on the Ventcell-Freidlin quasi-potential function. Appl. Math. Optim. 13 259-282.
5. DONSKER, M. D. and VARADHAN, S. R. S. (1975). On a variational formula for the principal eigenvalue for operators with maximum principle. Proc. Nat. Acad. Sci. U.S.A. 72 780-783.
6. DONSKER, M. D. and VARADHAN, S. R. S. (1975). Asymptotic evaluation of certain Wiener integrals for large time. In Functional Integration and Its Applications, Proceedings of the International Conference, London 15-33. Clarendon Press, Oxford.
7. DONSKER, M. D. and VARADHAN, S. R. S. (1975, 1976). Asymptotic evaluation of certain Markov process expectations for large: time. I, II, III, Comm. Pure Appl. Math. 28 1-45, 279-301; 29 389-461.
8. DONSKER, M. D. and VARADHAN, S. R. S. (1975, 1976). Asymptotic evaluation of certain Markov process expectations for large: time. I, II, III, Comm. Pure Appl. Math. 28 1-45, 279-301; 29 389-461.
9. DONSKER, M. D. and VARADHAN, S. R. S. (1976). On the principal eigenvalue of second-order elliptic differential operators. Commun. Pure Appl. Math. 24 595-621.
10. ∞, control problems. IEEE Trans. Automat. Control 34 831-847.
11. FLEMING, W. H. and HERNÁNDEZ-HERNÁNDEZ, D. (1995). Risk sensitive control of finite state machines on an infinite horizon I. Preprint.
12. FLEMING, W. H. and JAMES, M. R. (1992). Asymptotic series and exit time probabilities. Ann. Probab. 20 1369-1384.
13. ∞ norms for nonlinear systems. Math. Control, Signals, Systems 8 199-221.
14. FLEMING, W. H. and MCENEANEY, W. M. (1992). Risk sensitive optimal control and differential games. Proc. Conf. on Adaptive and Stochastic Control, Univ. Kansas. Lecture Notes in Control and Inform. Sci. 184 185-197. Springer, New York.
15. FLEMING, W. H. and MCENEANEY, W. M. (1995). Risk sensitive control on an infinite time horizon. SIAM J. Control Optim. 33 1881-1915.
16. FLEMING, W. H. and RISHEL, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, New York.
17. FLEMING, W. H., SHEU, S. J. and SONER, H. M. (1987). A remark on the large deviations of an ergodic Markov process. Stochastics 22 187-199.
18. FLEMING, W. H. and BONER, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.
19. FLEMING, W. H. and SOUGANIDIS, P. E. (1986). Asymptotic series and the method of vanishing viscosity. Indiana Univ. Math. J. 35 425-447.
20. ∞, Control Theory. Springer, New York.
21. FREIDLIN, M. I. and WENTZELL, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.
22. FRIEDMAN, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
23. GILBARG, D. and TRUDINGER, N. (1985). Elliptic Differential Equations of Second Order, 2nd ed. Springer, New York.
24. GLOVER, K. (1989). Minimum entropy and risk-sensitive control: the continuous time case. Proc. 28th IEEE Conf. on Decision and Control, Tampa, Florida 388-391.
25. HOLLAND, C. J. (1977). A new energy characterization of the smallest eigenvalue of the Schrödinger equation. Comm. Pure Appl. Math. 3 755-765.
26. ISHII, H., NAGAI, H. and TERAMOTO, F. (1996). A singular limit on risk sensitive control and semiclassical analysis. Proc. Seventh Japan-Russia Symp. on Probab. Theory and Math. Statist. World Scientific, Singapore.
27. JONA-LASINIO, G., MARTINELLI, F. and SCOPPOLA, E. (1981). New approach to the semiclassical limit of quantum mechanics. Comm. Math. Phys. 80 223-254.
28. KARATZAS, I. (1980). On a stochastic representation for the principal eigenvalue of a second order differential equation. Stochastics 3 305-321.
29. KARATZAS, I. and SHREVE, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.
30. KIFER, Yu. I. (1976). On the asymptotics of the transition density of processes with small diffusion. Theory Probab. Appl. 21 513-522.
31. KRYLOV, N. V. (1980). Controlled Diffusion Processes. Springer, New York.
32. KUSUOKA, S. and STROOCK, D. W. (1982). Applications of the Malliavin Calculus I. Taniguchi Symposium on Stochastic Analysis, Katata and Kyoto 271-306.
33. LI, P. and YAU, S. T. (1986). On the parabolic kernel of the Schrodinger operator. Acta Math. 156 153-201.
34. ∞ control. Preprint.
35. MIKAMI, T. (1988). Asymptotic expansions of the invariant density of a Markov process with a small parameter. Ann. Inst. H. Poincaré 24 403-424.
36. REED, M. and SIMON, B. (1978). Methods of Modern Mathematical Physics 4. Academic Press, New York.
37. SHEU, S.-J. (1984). Asymptotic expansion for the transition density of a diffusion Markov process with small diffusion. Stochastics 13 131-163.
38. SHEU, S.-J. (1984). Stochastic control and principal eigenvalue. Stochastics 11 191-211.
39. SHEU, S.-J. ( 1986). Asymptotic behavior of invariant density of diffusion Markov process with small diffusion. SIAM J. Math. Anal. 17 451-460.
40. SIMON, B. (1983). Semiclassical analysis of low lying eigenvalues I: nondegenerate minima: asymptotic expansion. Ann. Inst. H. Poincaré 38 297-307.
41. STROOCK, D. W. (1981). The Malliavin calculus and its application to second order parabolic differential equations. I. Math. Systems Theory 14 25-65.
42. STROOCK, D. W. (1984). An Introduction to the Theory of Large Deviations. Springer, New York.
43. STROOCK, D. W. and VARADHAN, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, New York.
44. VARADHAN, S. R. S. (1967). On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 431-455.
45. WATANABE, S. (1988). Generalized Wiener functionals and their applications. Lecture Notes in Math. 1299 541-548. Springer, Berlin.
46. WILLEMS, J. C. (1971). Least-squares stationary optimal control and the algebraic Riccati equation. IEEE Trans, Automat. Control 16 621-634.