A singular perturbation approach to first passage times for Markov jump processes. (English) Zbl 0629.60094
We introduce singular perturbation methods for constructing asymptotic approximations to the mean first passage time for Markov jump processes. Our methods are applied directly to the integral equation for the mean first passage time and do not involve the use of diffusion appoximations. An absorbing interval condition is used to properly account for the possible jumps of the process over the boundary which leads to a Wiener- Hopf problem in the neighborhood of the boundary. A model of unimolecular dissociation is considered to illustrate our methods.
MSC:
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
Keywords:
master equation; singular perturbation; asymptotic expansion; first passage time for Markov jump processes; diffusion appoximations; Wiener- Hopf problem; unimolecular dissociationReferences:
| [1] | B. J. Matkowsky and Z. Schuss,Bull. Am. Math. Soc. 82:321-324 (1976). · Zbl 0336.35010 · doi:10.1090/S0002-9904-1976-14041-4 |
| [2] | B. J. Matkowsky and Z. Schuss,SIAM J. Appl. Math. 33:365-382 (1977). · Zbl 0369.60071 · doi:10.1137/0133024 |
| [3] | Z. Schuss and B. J. Matkowsky,SIAM J. Appl. Math. 36:604-623 (1979). · Zbl 0406.60071 · doi:10.1137/0136043 |
| [4] | B. J. Matkowsky, Singular perturbations, stochastic differential equations and applications, inSingular Perturbations and Asymptotics, R. E. Meyer and S. V. Parter, eds. (Academic Press, New York, 1980). · Zbl 0494.60052 |
| [5] | B. J. Matkowsky and Z. Schuss,SIAM J. Appl. Math. 40:242-254 (1981). · Zbl 0477.60057 · doi:10.1137/0140020 |
| [6] | Z. Schuss,Theory and Applications of Stochastic Differential Equations (Wiley, New York, 1980). · Zbl 0439.60002 |
| [7] | Z. Schuss,SIAM Rev. 22:119-155 (1980). · Zbl 0436.60045 · doi:10.1137/1022024 |
| [8] | B. J. Matkowsky and Z. Schuss, Kramers’ diffusion and diffusion across characteristic boundaries, inTheory and Applications of Singular Perturbations, Conf. Proceedings, Oberwolfach, W. Eckhaus and E. M. de Jager, eds. (Springer Lecture Notes in Mathematics, No. 942, Springer, Berlin, 1981), pp. 318-345. |
| [9] | B. J. Matkowsky and Z. Schuss,SIAM J. Appl. Math. 42:822-834 (1981). · Zbl 0495.60078 · doi:10.1137/0142057 |
| [10] | B. J. Matkowsky, Z. Schuss, and E. Ben-Jacob,SIAM J. Appl. Math. 42:835-849 (1982). · Zbl 0494.76090 · doi:10.1137/0142058 |
| [11] | E. Ben-Jacob, D. Bergman, B. J. Matkowsky, and Z. Schuss,Phys. Rev. A 26:2805-2816 (1982). · doi:10.1103/PhysRevA.26.2805 |
| [12] | B. J. Matkowsky and Z. Schuss, Dynamical systems driven by small white noise: Asymptotic analysis and applications, inLecture Notes in Mathematics No. 985?Asymptotic Analysis II, F. ver Hulst, ed. (Springer-Verlag, Berlin, 1983), pp. 2-34. |
| [13] | B. J. Matkowsky, Z. Schuss, and C. Tier,SIAM J. Appl. Math. 43:673-695 (1983). · Zbl 0519.60055 · doi:10.1137/0143046 |
| [14] | E. Ben-Jacob, D. J. Bergman, B. J. Matkowsky and Z. Schuss, Thermal and shot noise effects on nonlinear oscillators, in Fifth International Conference on Collective Phenomena, J. Lebowitz, ed. (New York Academy of Sciences, New York, 1983), pp. 323-338. |
| [15] | B. J. Matkowsky and Z. Schuss,Phys. Lett. A95:213-215 (1983). |
| [16] | E. Ben-Jacob, D. J. Bergman, B. J. Matkowsky, and Z. Schuss,Phys. Lett. A99 (6,7):343-347 (1983). |
| [17] | E. Ben-Jacob, D. J. Bergman, Y. Imry, B. J. Matkowsky, and Z. Schuss,Appl. Phys. Lett. 42:1045-1047 (1983). · doi:10.1063/1.93837 |
| [18] | E. Ben-Jacob, D. J. Bergman, Y. Imry, B. J. Matkowsky, and Z. Schuss,J. Appl. Phys. 54:6533-6542 (1983). · doi:10.1063/1.331885 |
| [19] | B. J. Matkowsky, Z. Schuss, and C. Tier,J. Stat. Phys. 35:443-456 (1984). · doi:10.1007/BF01014395 |
| [20] | E. Ben-Jacob, D. J. Bergman, B. J. Matkowsky, and Z. Schuss, Noise-induced transitions in multi-stable systems, inFluctuations and Sensitivity in Nonequilibrium Systems?Proceedings of an International Conference, U. of Texas, March 1984, W. Horsthemke and D. K. Kondepudi, eds. (Springer Proceedings in Physics, No. 1, Springer, New York, 1984), pp. 79-94. |
| [21] | Z. Schuss, C. Tier, and B. J. Matkowsky,SIAM J. Appl. Math. 45:843-854 (1985). · Zbl 0576.60046 · doi:10.1137/0145050 |
| [22] | M. Dygas, B. J. Matkowsky, and Z. Schuss,SIAM J. Appl. Math. (to appear). |
| [23] | M. Dygas, B. J. Matkowsky, and Z. Schuss,J. Chem. Phys. 83:597-600 (1985). · doi:10.1063/1.449526 |
| [24] | B. J. Matkowsky, Z. Schuss, C. Knessl, C. Tier, and M. Mangel,Phys. Rev. A 29:3359-3369 (1984). · doi:10.1103/PhysRevA.29.3359 |
| [25] | C. Knessl, M. Mangel, B. J. Matkowsky, Z. Schuss, and C. Tier,J. Chem. Phys. 81:1285-1295. |
| [26] | B. J. Matkowsky, Z. Schuss, C. Knessl, C. Tier, and M. Mangel, First passage times governed by master equations, inFluctuations and Sensitivity in Nonequilibrium Systems?Proceedings of an International Conference, U. of Texas, March 1984, W. Horsthemke and D. K. Kondepudi, eds. (Springer Proceedings in Physics, No. 1, 19-36, 1984). · Zbl 0568.60071 |
| [27] | C. Knessl, B. J. Matkowsky, Z. Schuss, and C. Tier,SIAM J. Appl. Math. (to be published). |
| [28] | G. H. Weiss and A. Szabo,Physica 119A:569-579 (1983). |
| [29] | N. G. Van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981). · Zbl 0511.60038 |
| [30] | R. Kubo, K. Matsuo, and K. Kitahara,J. Stat. Phys. 9:51-96 (1973). · doi:10.1007/BF01016797 |
| [31] | P. Hanggi, H. Grabert, P. Talkner, and H. Thomas,Phys. Rev. A 29:371-378 (1984). · doi:10.1103/PhysRevA.29.371 |
| [32] | P. Hanggi and H. Thomas,Phys. Rep. 88C:207-319 (1982). · doi:10.1016/0370-1573(82)90045-X |
| [33] | N. G. Van Kampen and I. Oppenheim,J. Math. Phys. 13:842-849 (1972). · Zbl 0243.60041 · doi:10.1063/1.1666061 |
| [34] | W. Feller,Trans. Am. Math. Soc. 77:1-31 (1954). · doi:10.1090/S0002-9947-1954-0063607-6 |
| [35] | C. Knessl, B. J. Matkowsky, Z. Schuss, and C. Tier,SIAM J. Appl. Math. (to be published). |
| [36] | J. Troe,J. Chem. Phys. 66:4745-4757 (1977). · doi:10.1063/1.433837 |
| [37] | B. Carmeli and A. Nitzan,J. Chem. Phys. 76:5321-5333 (1982). · doi:10.1063/1.442930 |
| [38] | B. Noble,Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (Pergamon Press/MacMillan, London, 1958). · Zbl 0082.32101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
