Theory and applications of stochastic processes an analytical approach

Applied Mathematical Sciences (Switzerland)

Volumn 170, Issue , 2010, Pages 1-458

  Schuss, Zeev ^a  

^a TEL AVIV UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84901319201     PISSN: 00665452     EISSN: 2196968X     Source Type: Book Series    
DOI: 10.1007/978-1-4419-1605-1_1     Document Type: Chapter

References (254)

1. Abramowitz, M. and I. A. Stegun. Handbook of Mathematical Functions. Dover, New York, 1972.
2. Adelman, S.A. and J.D. Doll. “Generalized Langevin equation approach for atom/solid surface scattering: Collinear atom/harmonic chain model." J. Chem. Phys., 61, 4242–4245 (1974).
3. Agmon, N. “Diffusion with back reaction." J. Chem. Phys., 81, 2811–2817 (1984).
4. Allen, M.P. and D.J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, 1991.
5. Arnold, L. Random Dynamical Systems. Springer-Verlag, New York, 1998.
6. Arnold, L., G. Papanicolaou, and V. Wihstutz. “Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications." SIAM J. Appl. Math., 46, 427–450 (1986).
7. Asmussen, S., P. Glynn, and J. Pitman. “Discretization error in simulation of one-dimensional reflecting Brownian motion." Ann. Appl. Prob., 5 (4), 875– 896 (1995).
8. Bahadur, R.P. and R. Ranga Rao. “On deviations of the sample mean." Ann. Math. Stat., 31, 1015–1027 (1960).
9. Barndorf-Nielsen, O. and D.R. Cox. “Edgeworth and saddle point approximations with statistical applications." J. Royal Statist. Soc. B, 41 (3), 279–312 (1979).
10. Batsilas, L., A.M. Berezhkovskii, and S.Y. Shvartsman. “Stochastic model of autocrine and paracrine signals in cell culture assays." Biophys. J., 85, 3659Ű-3665 (2003).
11. Beccaria, M., G. Curci, and A. Vicere. “Numerical solutions of first-exit-time problems." Phys. Rev E., 48 (2), 1539–1546 (1993).
12. Belch, A.C. and M. Berkowitz. “Molecular dynamics simulations of tips2 water restricted by a spherical hydrophobic boundary." Chem. Phys. Lett., 113, 278–282 (1985).
13. Bellman, R., J. Bentsman, and S.M. Meerkov. “Nonlinear systems with fast parametric oscillations." J. Math. Anal. Appl., 97, 572–589 (1983).
14. Belopolskaya, Y.I. and Y.L. Dalecky. Stochastic Equations and Differential Geometry. Kluwer, Dordrecht, 1990.
15. Ben-Jacob, E., D. Bergman, and Z. Schuss. “Thermal fluctuations and the lifetime of the nonequilibrium steady-state in a hysteretic Josephson junction." Phys. Rev. B, Rapid Comm., 25, 519–522 (1982).
16. Ben-Jacob, E., D. Bergman, B.J. Matkowsky, and Z. Schuss. “Lifetime of oscillatory steady-states." Phys. Rev. A, 26, 2805–2816 (1982).
17. Ben-Jacob, E., D.J. Bergman, B.J. Matkowsky, and Z. Schuss. “Noise induced transitions in multi-stable systems," in Fluctuations and Sensitivity in Nonequilibrium Systems, Proceedings of an International Conference, University of Texas, Austin, Texas, March 12–16. Springer Proceedings in Physics 1, 1984.
18. Ben-Jacob, E., D.J. Bergman, B.J. Matkowsky, and Z. Schuss. “Master-equation approach to shot noise in Josephson junctions." Phys. Rev. B, 34 (3), 1572–1581 (1986).
19. Ben-Jacob, E., D.J. Bergman, Y. Imry, B.J. Matkowsky, and Z. Schuss. “Thermal activation from the fluxoid and the voltage states of dc SQUIDs. J. Appl. Phys., 54, 6533–6542 (1983).
20. Bender, C.M. and S.A. Orszag. Advanced Mathematical Methods for Scien-tists and Engineers. McGraw-Hill, New York, 1978.
21. Berezhkovskii, A.M., Y.A. Makhnovskii, M.I. Monine, V.Yu. Zitserman, and S.Y. Shvartsman. “Boundary homogenization for trapping by patchy surfaces." J. Chem. Phys., 121 (22), 11390–11394 (2004).
22. Berkowitz, M. and J.A. McCammon. “Molecular dynamics with stochastic boundary conditions." Chem. Phys. Lett., 90, 215–217 (1982).
23. Berne, B.J. and R. Pecora. Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics. Wiley-Interscience New York, 1976.
24. Berry, R.S.,S. Rice, and J. Ross. Physical Chemistry. 2nd edition. Oxford University Press, Oxford. 2000.
25. Billingsley, P. Convergence of Probability Measures. J. Wiley and Sons, New York, 1968.
26. Bobrovsky, B.Z. and O. Zeitouni. “Some results on the problem of exit from a domain." Stoch. Process. Appl., 41 (2), 241–256 (1992).
27. Bobrovsky, B.Z. and Z. Schuss. “A singular perturbation method for the computation of the mean first passage time in a nonlinear filter." SIAM J. Appl. Math., 42 (1), 174–187 (1982).
28. Bossy, M., E. Gobet, and D. Talay. “A symmetrized Euler scheme for an efficient approximation of reflected diffusions." J. Appl. Prob., 41, 877–889 (2004).
29. Brooks, C.L., III and M. Karplus. “Deformable stochastic boundaries in molecular dynamics." J. Chem. Phys., 79, 6312 (1983).
30. Brush, S.G. The Kind of Motion We Call Heat, I, II. North Holland, Amsterdam, 1986.
31. Bryan, G.H. “Diffusion with back reaction." Proc. Camb. Phil. Soc., 7, 246, (1891).
32. Bucklew, J.A. Large Deviation Techniques in Decision, Simulation and Esti-mation. Wiley, New York, 1990.
33. Büttiker, M., E.P. Harris, and R. Landauer. “Thermal activation in extremely underdamped Josephson-junction circuits." Phys. Rev. B, 28, 1268 (1979).
34. Cameron, R.H. and W.T. Martin. “The transformation of Wiener integrals by nonlinear transformations." Trans. Amer. Math. Soc., 69, 253Ű–283 (1949).
35. Carmeli, B. and A. Nitzan. “First passage times and the kinetics of unimolec-ular dissociation." J. Chem. Phys., 76, 5321–5333 (1982).
36. Carmeli, B. and A. Nitzan. “Non-Markovian theory of activated rate processes V, external periodic forces in the low friction limit." Phys. Rev. A, 32, 2439–2454 (1985).
37. Carslaw, H.S. and J. C. Jaeger. Conduction of Heat in Solids. 2nd edition. Oxford University Press, Oxford. 1959.
38. Chandrasekhar, S. “Stochastic problems in physics and astronomy." Rev. Mod. Phys., 15, 2–89 (1943).
39. Christiano, R. and P. Silvestrini. “Decay of the running state in Josephson junctions: Preliminary experimental results." Phys. Lett. A, 133 (2), 347 (1988).
40. Cochran, J.A. The Analysis of Linear Integral Equations. McGraw-Hill, New York, 1972.
41. Coddington, E. and N. Levinson. Ordinary Differential Equations. McGraw-Hill, New York, 1955.
42. Coffey, W.T., Yu.P. Kalmykov, and J.T. Waldron. The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering. 2nd edition. Series in Contemporary Chemical Physics Vol. 14. World Scientific, New York, 2004.
43. Collins, F.C. and G.E. Kimball. “Random diffusion-controlled reaction rates." J. Colloid Sci., 4, 425–437 (1949).
44. Costantini, C., B. Pacchiarotti, and F. Sartoretto. “Numerical approximation for functionals of reflecting diffusion processes." SIAM J. Appl. Math., 58 (1), 73–102 (1998).
45. Courant, R. and D. Hilbert. Methods of Mathematical Physics. Wiley-Interscience, New York, 1989.
46. Cox, D.R. and H.D. Miller. The Theory of Stochastic Processes. Chapman and Hall, London, 1984.
47. Cramér, H. and M.R. Leadbetter. Stationary and Related Stochastic Pro-cesses. Wiley, New York, 1967.
48. Daniels, H.E. “Saddle point approximations in statistics." Ann. Math. Stat., 25, 631–650 (1954).
49. Day, M.V. “Conditional exits for small noise diffusions." Ann. Prob., 20, 1385–1419 (1992).
50. Day, M.V. “Boundary local time and small parameter exit problems with characteristic boundaries." SIAM J. Math. Anal., 20, 222–248 (1989).
51. Day, M.V. Large deviations results for the exit problem with characteristic boundary. J. Math. Anal. Appl., 147, 134–153 (1990).
52. Day, M.V. “Some phenomena of the characteristic boundary exit problem," in Diffusion Processes and Related Problems in Analysis. Birkhäuser, Basel 1990.
53. De Finetti, B. Theory of Probability. Wiley, New York, 3rd edition. 1993.
54. Dembo, A. and O. Zeitouni. Large Deviations Techniques and Applications. Jones and Bartlett, Boston. 1993.
55. Deuschel, J.D. and D.W. Stroock. Large Deviations. Academic Press, 1989.
56. Devinatz, A. and A. Friedman. “The asymptotic behavior of the solution of a singularly perturbed Dirichlet problem." Indiana Univ. Math. J., 27 (3), 527–537 (1978).
57. Doob, J.L. Stochastic Processes. Wiley, New York, 1953.
58. Doob, J.L. Measure Theory. Springer Verlag, New York, 1994.
59. Doucet, A., N.D. Freitas, and N. Gordon. Sequential Monte-Carlo Methods in Practice. Springer, Berlin, 2001.
60. Dvoretzky, A., P. Erdös, and S. Kakutani. “Nonincreasing everywhere of the Brownian motion process," in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability II, University of California Press, Bekeley. 1961.
61. Dygas, M.M., B.J. Matkowsky, and Z. Schuss. “A singular perturbation approach to non-Markovian escape rate problems." SIAM J. Appl. Math., 46 (2), 265–298 (1986).
62. Dynkin, E.B., T. Kovary, and D. E. Brown. Theory of Markov Processes. Dover, New York, 2006.
63. Einstein, A. Investigations on the Theory of the Brownian Movement. translated and reprinted by Dover, New York 1956.
64. Eisenberg, R.S. “Ionic channels in biological membranes. Electrostatic analysis of a natural nanotube." Contemp. Phys., 39 (6), 447–466 (1998).
65. Eisenberg, R.S. “From structure to function in open ionic channels." J. Membrane Biol., 171, 1–24 (1999).
66. Eisenberg, R.S., M.M. Kłosek, and Z. Schuss. “Diffusion as a chemical reac-tion: Stochastic trajectories between fixed concentrations." J. Chem. Phys., 102 (4), 1767–1780 (1995).
67. Eliezer, B. and Z. Schuss. “Stochastic (white noise) analysis of resonant absorption in laser generated plasma." Phys. Lett. A, 70 (4), 307–310 (1979).
68. Ellis, R.S. Entropy, Large Deviations and Statistical Mechanics. Springer-Verlag, New York, 1985.
69. Erban, R. and J. Chapman. “Reactive boundary conditions for stochastic simulations of reactionŰdiffusion processes." Phys. Biol., 4, 16Ű-28 (2007).
70. Ethier, S.N. and T.G. Kurtz. Markov Processes: Characterization and Con-vergence. 2nd edition. Wiley Series in Probability and Statistics, New York, 2005.
71. Feller, W. “Diffusion processes in one dimension." Trans. AMS, 77 (1), 1–31 (1954).
72. Feller, W. An Introduction to Probability Theory and Its Applications, volumes I, II. 3rd edition. John Wiley & Sons, New York, 1968.
73. Fichera, G. “On a unified theory of boundary value problems for-elliptic-parabolic equations of second order." in Boundary Value Problems in Differential Equations, University of Wisconsin Press, Madison. 1960.
74. Freidlin, M. Markov Processes and Differential Equations. Birkhäuser, Boston, 2002.
75. Freidlin, M. “Some remarks on the Smoluchowski–Kramers approximation." J. Stat. Phys., 117 (3/4), 617–654 (2004).
76. Freidlin, M.A. Functional Integration and Partial Differential Equations. Princeton University Press, Princeton, NJ, 1985.
77. Freidlin, M.A. and A.D. Wentzell. Random Perturbations of Dynamical Systems. Springer-Verlag, New York, 1984.
78. Friedman, A. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ, 1964.
79. Friedman, A. Foundations of Modern Analysis. Dover, New York, 1983.
80. Friedman, A. Stochastic Differential Equations and Applications. Dover, New York, 2007.
81. Gantmacher, F.R. The Theory of Matrices. American Mathematical Society, Providenc, RI, 1998.
82. Gardiner, C.W. Handbook of Stochastic Methods. 2nd edition. Springer Ver-lag, New York, 1985.
83. Ghoniem, A.F. and F.S. Sherman. “Grid free simulation of diffusion using random walk methods." J. Comp. Phys., 61, 1–37 (1985).
84. Gihman, I.I. and A.V. Skorohod. Stochastic Differential Equations. Springer-Verlag, Berlin, 1972.
85. Gihman, I.I. and A.V. Skorohod. The Theory of Stochastic Processes, Vols. I, II, III. Springer-Verlag, Berlin, 1975.
86. Girsanov, I.V. “On transforming a certain class of stochastic processes by absolutely continuous substitution of measures." Theory Probab. Appl., 5, 285–301 (1960).
87. Glasstone, S., K.J. Laidler, and H. Eyring. The Theory of Rate Processes. McGraw-Hill, New York, 1941.
88. Gobet, E. “Euler schemes and half-space approximation for the simulation of diffusion in a domain." ESAIM Probab. Statist., 5, 261–297 (2001).
89. Goodrich, F.C. “Random walk with semiadsorbing barrier." J. Chem. Phys., 22, 588–594 (1954).
90. Graham, R. and T. Tel. “Existence of a potential for dissipative dynamical systems." Phys. Rev. Lett., 52, 9–12 (1984).
91. Grote, F.R. and J.T. Hynes. “The stable states picture of chemical reactions II: Rate constants for condensed and gas phase reaction models." J. Chem Phys., 73, 2715–2732 (1980).
92. Grote, F.R. and J.T. Hynes. “Reactive modes in condensed phase reactions." J. Chem Phys., 74, 4465–4475 (1981).
93. Grote, F.R. and J.T. Hynes. “Saddle point model for atom transfer reactions in solution." J. Chem Phys., 75, 2191–2198 (1981).
94. Hagan, P.S., C.R. Doering, and C.D. Levermore. “Bistability driven by weakly colored Gaussian noise: The Fokker–Planck boundary layer and mean first-passage times." Phys. Rev. Lett., 59, 2129Ű–2132 (1987).
95. Hagan, P.S., C.R. Doering, and C.D. Levermore. “The distribution of exit times for weakly colored noise." J. Stat. Phys., 54 (5/6), 1321–1352 (1989).
96. Hagan, P.S., C.R. Doering, and C.D. Levermore. “Mean exit times for particles driven by weakly colored noise." SIAM J. Appl. Math., 49 (5), 1480– 1513 (1989).
97. Hale, J.K. Ordinary Differential Equations. Wiley, New York, 1969.
98. Hänggi, P. Activated Barrier Crossing: Applications in Physics, Chemistry and Biology. World Scientific, Singapore, 1993.
99. Hänggi, P. and P. Talkner. New Trends in Kramers’ Reaction Rate Theory. Kluwer, Dordrecht 1995.
100. Hänggi, P., P. Talkner, and M. Borkovec. “50 years after Kramers." Rev. Mod. Phys., 62, 251–341 (1990).
101. Hida, T. Brownian Motion. Springer, New York, 1980.
102. Hille, B. Ionic Channels of Excitable Membranes. 3rd edition. Sinauer, Sun-derland, 2001.
103. Honerkamp, J. Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis. VCH, New York, 1994.
104. Im, B., S. Seefeld, and B. Roux. “A grand canonical Monte-Carlo-Brownian dynamics algorithm for simulating ion channels." Biophys. J., 79, 788–801 (2000).
105. Iscoe, I., P. Ney, and E. Nummelin. “Large deviations of uniformly recurrent Markov additive processes." Adv. Appl. Math., 6, 373–412 (1985).
106. Itô, K. and H.P. McKean, Jr. Diffusion Processes and Their Sample Paths. Classics in Mathematics. Springer-Verlag, New York, 1996.
107. Jackson, J.D. Classical Electrodymnics. 2nd edition. Wiley, New York, 1975.
108. Jazwinski, A.H. Stochastic Processes and Filtering Theory. Academic Press, New York, 1970.
109. Jensen, J.L. “Saddle point expansions for sums of Markov dependent variables on a continuous state space." Probab. Theor. Related Fields, 89, 181– 199 (1991).
110. Jensen, J.L. Saddlepoint Approximations. Oxford Statistical Science Series, Oxford, 1995.
111. Johnson, J.B. “Thermal agitation of electricity in conductors." Phys. Rev., 32, 97–109 (1928).
112. Kamienomostskaya, S. “On equations of elliptic type and parabolic type with a small parameter in the highest derivative." Mat. Sbornik, 31 (73), 703–708 (1952).
113. Kamienomostskaya, S. “The first boundary value problem for elliptic equations containing a small parameter." Izv. Akad. Nauk U.S.S.R., 27 (6), 935– 360 (1955).
114. Kamin, S. “On elliptic perturbation of a first order operator with a singular point of attracting type." Indiana Univ. Math. J., 27 (6), 935–951 (1978).
115. Karatzas, I. and S.E. Shreve. Brownian Motions and Stochastic Calculus. 2nd edition. Springer Graduate Texts in Mathematics 113, New York, 1991.
116. Karlin, S. and H.M. Taylor. A First Course in Stochastic Processes, Vol. I, 2nd edition. Academic Press, New York, 1975.
117. Karlin, S. and H.M. Taylor. A Second Course in Stochastic Processes, Vol. II, 2nd edition. Academic Press, New York, 1981.
118. Katz, A. Reliability of Elastic Structures Driven by Random Loads. Ph.D. dissertation, Department of Mathematics, Tel-Aviv University, Israel, 1985.
119. Katz, A. and Z. Schuss. “Reliability of elastic structures driven by random loads." SIAM J. Appl. Math., 45 (3), 383–402 (1985).
120. Keller, J.B. and D.W. McLaughlin. “The Feynman integral." Amer. Math. Month., 82 (5), 451–576 (1975).
121. Kevorkian, J. and J.D. Cole. Perturbation Methods in Applied Mathematics. Applied Mathematical Sciences. Springer-Verlag, Berlin and Heidelberg, 1985.
122. Khasminskii, R.Z. and A.M. Il’in. “On equations of Brownian motion." Theor. Probab. Appl., 9, 421–444 (1964).
123. Kim, S. and I. Oppenheim. “Molecular theory of Brownian motion in external fields." Physica, 57, 469–482 (1972).
124. Kłosek-Dygas, M.M., B.J. Matkowsky, and Z. Schuss. “Colored noise in dynamical systems." SIAM J. Appl. Math., 48 (2), 425–441 (1988).
125. Kłosek-Dygas, M.M., B.J. Matkowsky, and Z. Schuss. “A first passage time approach to stochastic stability of nonlinear oscillators." Phys. Lett. A, 130 (1), 11–18 (1988).
126. Kłosek-Dygas, M.M., B.J. Matkowsky, and Z. Schuss. “Stochastic stability of nonlinear oscillators." SIAM J. Appl. Math., 48 (5), 1115–1127 (1988).
127. Kłosek, M.M. “Half-range expansions for analysis for Langevin dynamics in the high friction limit with a singular boundary condition: Non characteristic case." J. Stat. Phys., 79 (1/2), 313–345 (1995).
128. Kłosek, M.M., and P.S. Hagan. “Colored noise and a characteristic level crossing problem." J. Math. Phys., 39, 931–953 (1998).
129. Kleinert, H. Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics. World Scientific, New York, 1994.
130. Kleinrock, L. Queueing Systems, Vols. I, II. Wiley, New York,, 1975, 1976.
131. Kloeden, P.E. “The systematic derivation of higher order numerical schemes for stochastic differential equations." Milan J. Math., 70, 187Ű-207 (2002).
132. Kloeden, P.E. and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York, 1992.
133. Knessl, C., B.J. Matkowsky, Z. Schuss, and C. Tier. “An asymptotic theory of large deviations for Markov jump processes." SIAM J. Appl. Math., 45, 1006–1028 (1985).
134. Knessl, C., B.J. Matkowsky, Z. Schuss, and C. Tier. “Asymptotic analysis of a state dependent M/G/1 queueing system." SIAM J. Appl. Math., 46 (3), 483–505 (1986).
135. Knessl, C., B.J. Matkowsky, Z. Schuss, and C. Tier. “Boundary behavior of diffusion approximations to Markov jump processes." J. Stat. Phys., 45, 245– 266 (1986).
136. Knessl, C., B.J. Matkowsky, Z. Schuss, and C. Tier. “On the performance of state dependent single server queues." SIAM J. Appl. Math., 46 (4), 657–697 (1986).
137. Knessl, C., B.J. Matkowsky, Z. Schuss, and C. Tier. “A singular perturbations approach to first passage times for Markov jump processes." J. Stat. Phys., 42, 169–184 (1986).
138. Knessl, C., B.J. Matkowsky, Z. Schuss, and C. Tier. “System crash in finite capacity M/G/1 queue." Commun. Stat.-Stochastic Models, 2 (2), 171–201 (1986).
139. Korkotian, E. and M. Segal. “Spike-associated fast contraction of dendritic spines in cultured hippocampal neurons." Neuron, 30 (3), 751–758 (2001).
140. Kramers, H.A. “Brownian motion in field of force and diffusion model of chemical reaction." Physica, 7, 284–304 (1940).
141. Kubo, R. “Statistical-mechanical theory of irreversible processes, I." J. Phys. Soc. Japan, 12, 570–586 (1957).
142. Kupferman, R., M. Kaiser, Z. Schuss, and E. Ben-Jacob. “A WKB study of fluctuations and activation in nonequilibrium dissipative steady-states." Phys. Rev. A, 45 (2), 745–756 (1992).
143. Kushner, H.J. “A cautionary note on the use of singular perturbation methods for small-noise models." Stochastics, 6, 116-Ű120 (1982).
144. Kushner, H.J. and G. Yin. Stochastic Approximation Algorithms and Applications, revised 2nd edition. Springer-Verlag, Berlin and New York, 2003.
145. Lamm, G. and K. Schulten. “Extended Brownian dynamics. II: reactive, non-linear diffusion." J. Chem. Phys., 78 (5), 2713–2734 (1983).
146. Landauer, R. in Self-Organizing Systems: The Emergence of Order. Edited by F.E. Yates, D.O. walter, and G.B. Yates. Plenum Press, New York, 1983.
147. Langevin, P. “Sur la théorie du mouvement Brownien." C.R. Paris, 146, 530– 533 (1908).
148. Lee, R.C.K. Optimal Estimation Identification and Control. MIT Press, Cambridge, MA, 1964.
149. Lépingle, D. “Euler scheme for reflected stochastic differential equations." Math. Comput. Simul., 38, 119–126 (1995).
150. Lévy, P. Le Mouvement Brownien. Mémorial des Sciences Mathématiques, Fasc. 76, Paris, 1954.
151. Li, G.W. and G.L. Blankenship. “Almost sure stability of linear stochastic systems with poisson process coefficients." SIAM J. Appl. Math., 46, 875– 911 (1986).
152. Lighthill, M.J. Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press, New York, 1958.
153. Liptser, R.S. and A.N. Shiryayev. Statistics of Random Processes, Vols. I, II. Springer-Verlag, New York, 1977.
154. Ludwig, D. “Persistence of dynamical systems under random perturbations." SIAM Rev., 17 (4), 605–640 (1975).
155. Maier, R. and D.L. Stein. “Effect of focusing and caustics on exit phenomena in systems lacking detailed balance." Phys. Rev. Lett., 71, 1783–1786 (1993).
156. Maier, R. and D.L. Stein. “Escape problem for irreversible systems." Phys. Rev. E, 48, 931–938 (1993).
157. Maier, R. and D.L. Stein. “Oscillatory behavior of the rate of escape through an unstable limit cycle." Phys. Rev. Lett., 77, 4860–4863 (1996).
158. Maier, R. and D.L. Stein. “A scaling theory of bifurcation in the symmetric weak-noise escape problem." J. Stat. Phys., 83, 291–357 (1996).
159. Maier, R. and D.L. Stein. “Limiting exit location distributions in the stochastic exit problem." SIAM J. Appl. Math., 57, 752–790 (1997).
160. Maier, R., D.G. Luchinsky, R. Mannella, P.V.E. McClintock, and D.L. Stein. “Experiments on critical phenomena in a noisy exit problem." Phys. Rev. Lett., 79, 3109–3112 (1997).
161. Mandl, P. Analytical Treatment of One-Dimensional Markov Processes. Springer Verlag, New York, 1968.
162. Mangel, M. and D. Ludwig. “Probability of extinction in a stochastic compe-tition." SIAM J.Appl. Math., 33 (2), 256 (1977).
163. Mannella, R. “Absorbing boundaries and optimal stopping in a stochastic differential equation." Phys. Lett. A., 254 (5), 257–262 (1999).
164. Mannella, R. “Integration of stochastic differential equations on a computer." Int. J. Mod. Phys. C, 13 (9), 1177–1194 (2002).
165. Marchewka, A. and Z. Schuss. “Path integral approach to the Schrödinger current." Phys. Rev. A, 61, 052107 (2000).
166. Marshall, T.W. and E.J. Watson. “A drop of ink falls from my pen … it comes to earth, I know not when." J. Phys. A, 18, 3531–3559 (1985).
167. Matkowsky, B.J. and Z. Schuss. “The exit problem for randomly perturbed dynamical systems." SIAM J. Appl. Math., 33, 365–382 (1977).
168. Matkowsky, B.J. and Z. Schuss. “Eigenvalues of the Fokker–Planck operator and the approach to equilibrium in potential fields." SIAM J. Appl. Math., 40, 242–252 (1981).
169. Matkowsky, B.J., Z. Schuss, and C. Tier. “Diffusion across characteristic boundaries with critical points." SIAM J. Appl. Math., 43, 673–695 (1983).
170. Mazur, P. and I. Oppenheim. “Molecular theory of Brownian motion." Physica, 50, 241–258 (1970).
171. McKean, H.P., Jr. Stochastic Integrals. Academic Press, New York, 1969.
172. Mel’nikov, V.I. and S.V. Meshkov. “Theory of activated rate processes: Exact solution of the Kramers problem." J. Chem. Phys., 85, 1018–1027 (1986).
173. Miller, W.H. “Quantum-mechanical transition state theory and a new semi-classical model for reaction rate constants." J. Chem. Phys., 61, 1823–1834 (1974).
174. Milshtein, G.N. “A method of second-order accuracy integration of stochastic differential equations." Theor. Probab. Appl., 23, 396–401 (1976).
175. Milshtein, G.N. Numerical Integration of Stochastic Differential Equations. Mathematics and Its Applications. Kluwer, Dordrecht, 1995.
176. Monine, M.I. and J.M. Haugh. “Reactions on cell membranes: Comparison of continuum theory and Brownian dynamics simulations." J. Chem. Phys., 123, 074908 (2005).
177. Mori, H. “Transport, collective motion, and Brownian motion." Prog. Theor. Phys., 33, 423–454 (1965).
178. Musiela, M. and M. Rutkowski. Martingale Methods in Financial Modeling. 2nd edition. Applications of Mathematics 36, Springer Verlag, New York, 1997.
179. Nadler, B., T. Naeh, and Z. Schuss. “The stationary arrival process of dif-fusing particles from a continuum to an absorbing boundary is Poissonian." SIAM J. Appl. Math., 62 (2), 433–447 (2002).
180. Nadler, B., T. Naeh, and Z. Schuss. “Connecting a discrete ionic simulation to a continuum." SIAM J. Appl. Math., 63 (3), 850–873 (2003).
181. Naeh, T. Simulation of Ionic Solutions. Ph.D. dissertation, Department of Mathematics. Tel-Aviv University, Israel, 2001.
182. Naeh, T., M.M. Kłosek, B. J. Matkowsky, and Z. Schuss. “Direct approach to the exit problem." SIAM J. Appl. Math., 50, 595–627 (1990).
183. Natanson, I.P. Theory of Functions of a Real Variable. Ungar, New York, 1961.
184. Nevel’son, M.B. and R.Z. Has’minski, Stochastic Approximation and Recur-sive Estimation, translated by Israel Program for Scientific Translations and B. Silver, AMS, Providence, RI, 1976.
185. Nitzan, A. Chemical Dynamics in Condensed Phases Relaxation, Transfer and Reactions in Condensed Molecular Systems. Graduate Texts edition. Oxford University Press, New York, 2006.
186. Nitzan, A. and Z. Schuss. “Multidimensional barrier crossing." in Activated Barrier Crossings, G.R. Fleming and P. Hänggi, Editors. World Scientific, Singapore, 1993.
187. Noble, B. Methods Based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations. AMS/Chelsea, Providence, RI, 1988.
188. Nonner, W., D. Gillespie, D. Henderson, and R.S. Eisenberg. “Ion accumu-lation in a biological calcium channel: Effects of solvent and confining pressure." J. Ph. Chem. B, 105, 6427–6436 (2001).
189. Nonner, W., D.P. Chen, and R.S. Eisenberg. “Anomalous mole fraction effect, electrostatics, and binding in ionic channels." Biophys. J., 74, 2327–2334 (1998).
190. Nonner, W., L. Catacuzzeno, and R.S. Eisenberg. “Binding and selectivity in l-type ca channels: A mean spherical approximation." Biophys. J., 79, 1976– 1992 (2000).
191. Nonner, W. and R.S. Eisenberg. “Ion permeation and glutamate residues linked by Poisson–Nernst–Planck theory in L-type calcium channels." Bio-phys. J., 75, 1287–1305 (1998).
192. Nyquist, H. “Thermal agitation of electric charge in conductors." Phys. Rev., 32, 110Ű113 (1928).
193. Øksendal, B. Stochastic Differential Equations. 5th edition. Springer, Berlin Heidelberg, 1998.
194. Olver, F.W.J. Asymptotics and Special Functions. Academic Press, New York, and London, 1974.
195. O’Malley, R.E., Jr. Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag, New York, 1991.
196. Oppenheim, I., K.E. Shuler, and G.H. Weiss. Stochastic Processes in Chemical Physics: The Master Equation. MIT Press, Cambridge MA, 1977.
197. Paley, R.N. and N. Wiener. Fourier Transforms in the Complex Plane. AMS Colloquium Publicaions, Vol 19, New York, 1934.
198. Paley, R.N., N. Wiener, and A. Zygmund. “Note on random functions." Math. Z., 37, 647–668 (1933).
199. Papoulis, A. Probability, Random Variables, and Stochastic Processes. 3rd edition. McGraw Hill, New York, 1991.
200. Perrin, J. “L’agitation moléculaire et le mouvement Brownien." C.R. Paris, 146, 967–970 (1908).
201. Peters, E.A.J.F. and Th.M.A.O.M. Barenbrug. “Efficient Brownian dynamics simulation of particles near walls. I: Reflecting and absorbing walls." Phys. Rev. E, 66, 056701 (2002).
202. Pinsky, R.G. “Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small punc-ture." J. Funct. Anal., 200 (1), 177–197 (2003).
203. Protter, M.H. and H.F. Weinberger. Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ, 1967.
204. Protter, P. Stochastic Integration and Differential Equations. Springer-Verlag, New York, 1992.
205. Ramo, S. “Currents induced by electron motion." Proc. IRE., 9, 584–585 (1939).
206. Risken, H. The Fokker–Planck Equation: Methods of Solutions and Applica-tions. 2nd edition. Springer-Verlag, New York, 1996.
207. Robert, C. and G. Casella. Monte-Carlo Statistical Methods. Springer, Berlin, 1999.
208. Rogers, L.C.G. and D. Williams. Diffusions, Markov Processes, and Martingales, Vols. I, II. Cambridge University Press, New York, 2000.
209. Rubin, R.J. “Statistical dynamics of simple cubic lattices. model for the study of Brownian motion." J. Math. Phys., 1, 309–318 (1960).
210. Ryter, D. and H. Meyr. “Theory of phase tracking systems of arbitrary order." IEEE Trans. Inf. Theor., IT-24, 1–7 (1978).
211. Schulman, L.S. Techniques and Applications of Path Integrals. Wiley, New York, 1981.
212. Schumaker, M. “Boundary conditions and trajectories of diffusion pro-cesses." J. Chem. Phys., 117, 2469–2473 (2002).
213. Schuss, Z. Theory and Applications of Stochastic Differential Equations. Wi-ley, New York, 1980.
214. Schuss, Z. Brownian Dynamics and Simulations. Preprint, 2009.
215. Schuss, Z. Optimal Phase Tracking. Preprint, 2009.
216. Schuss, Z. and A. Spivak. “Where is the exit point?" Chem. Phys., 235, 227– 242 (1998).
217. Schuss, Z. and A. Spivak. “The exit distribution on the stochastic separatrix in Kramers’ exit problem." SIAM J. Appl. Math., 62 (5), 1698–1711 (2002).
218. Schuss, Z. and B.J. Matkowsky. “The exit problem: A new approach to diffusion across potential barriers." SIAM J. Appl. Math., 35 (3), 604–623 (1979).
219. Schuss, Z., B. Nadler, and R.S. Eisenberg. “Derivation of PNP equations in bath and channel from a molecular model." Phys. Rev. E, 64, (2-3), 036116– 1–036116–14, 2001.
220. Shockley, W. “Currents to conductors induced by moving point charge." J. Appl. Phys., 9, 635–636 (1938).
221. Singer, A. Diffusion Theory of Ion Permeation Through Protein Channels of Biological Membranes. Ph.D. dissertation, Department of Mathematics, Tel-Aviv University, 2006.
222. Singer, A. and Z. Schuss. “Brownian simulations and unidirectional flux in diffusion." Phys. Rev. E, 71, 026115 (2005).
223. Singer, A., Z. Schuss, A. Osipov, and D. Holcman. “Partially absorbed diffu-sion." SIAM J. Appl. Math., 68 (3), 844–868 (2008).
224. Skorokhod, A.V. “Stochastic equations for diffusion processes in a bounded region." Theor. Probab. Appl., 6 (3), 264–274 (1961).
225. Slemrod, M. “Stabilization of bilinear control systems with application in nonconservative problems in elasticity." SIAM J. Control Opt., 16, 131–141 (1978).
226. Smoluchowski, M. von. “Studien über Molekularstatistik von Emulsionen und deren Zusammenhang mit der Brown’schen Bewegung." Wien. Ber., 123, 2381–2405 (1914).
227. Smoluchowski, M. von. Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen." Phys. Zeits., 17, 557 (1916).
228. Smoluchowski, M.R. von Smolan. “Zarys kinetycznej teorji ruchów Browna." Rozprawy Wydziału Przyrodniczego Akademii Umiejȩtności (Kraków), A46, 257–282 (1906).
229. Sneddon, I. Elements of Partial Differential Equations. McGraw-Hill International (Mathematics Series), 1985.
230. Song, Y., Y. Zhang, T. Shen, C.L. Bajaj, J.A. McCammon, and N.A. Baker. “Finite element solution of the steady-state Smoluchowski equation for rate constant calculations." Biophys. J., 86, 2017–2029 (2004).
231. Spitzer, F. “Some theorems concerning 2-dimensional Brownian motion." Trans. AMS 87, 187–197 (1958).
232. Stratonovich, R. Topics in the Theory of Random Noise. Mathematics and Its Applications, vols I, II. Gordon and Breach, New York, 1963.
233. Stroock, D.W. An Introduction to Markov Processes. Springer Graduate Texts in Mathematics, New York, 2005.
234. Stroock, D.W. and S.R.S. Varadhan. Multidimensional Diffusion Processes. Springer-Verlag, New York, Grundlehren der Mathematischen Wis-senschaften 233, 1979.
235. Sutherland, W. “A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin." Philos. Mag., 9 (5), 781–785 (1905).
236. Svedberg, T. “Über die Eigenbewegung der Teilchen in kolloidalen Lösungen I." Z. Elektrochem. Angewandte Phys. Chem., 12, 853–860 (1906).
237. Svedberg, T. Nachweis der von der kinetischen Theorie Geforderten Bewe-gung gelöster Moleküle." Zeits. f. physik Chemie, 74, 738 (1910).
238. Szymczak, P. and A.J.C. Ladd. “Boundary conditions for stochastic solutions of the convection-diffusion equation." Phys. Rev. E, 68, 036704 (2003).
239. Tai, K., S.D. Bond, H.R. MacMillan, N.A. Baker, M.J. Holst, and J.A. Mc-Cammon. “Finite element simulations of acetylcholine diffusion in neuro-muscular junctions." Biophys. J., 84, 2234Ű–2241 (2003).
240. Troe, J. “Theory of thermal unimolecular reactions at low pressures: I. Solutions of the master equation." J. Chem. Phys., 66, 4745–4757 (1977).
241. Üstünel, A.S. and M. Zakai. Transformation of Measure on Wiener Space. Springer, Berlin, 2000.
242. van Kampen, N.G. Stochastic Processes in Physics and Chemistry. 3rd edition. North-Holland Personal Library, JAI Press, Greenwich, CT, 1992.
243. Van Trees, H.L. Detection, Estimation and Modulation Theory, Vols. I, II, III. John Wiley, New York, 1970.
244. Viterbi, A.J. Principles of Coherent Communication. McGraw-Hill, New York, 1966.
245. Walker, J.A. “On the application of Lyapunov’s direct method to linear lumped-parameter elastic systems." J. Appl. Mech., 41, 278–284 (1974).
246. Westgren, A. “Die Veranderungsgeschwindigkeit der lokalen Teilchenkonzentration in Kolloiden Systemen, I." Arkiv for Matematik, Astronomi, och Fysik, 11 (14), 1–24 (1916).
247. Westgren, A. “Die Veranderungsgeschwindigkeit der lokalen Teilchenkonzentration in Kolloiden Systemen, II." Arkiv for Matematik, Astronomi, och Fysik, 13 (14), 1–18 (1918).
248. Wiener, N. “Differential space." J. Math. Phys., 2, 131–174 (1923).
249. Wilmott, P. J. Dewynne, and S. Howison. Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford, 1994.
250. Yu, A.W., G.P. Agrawal, and R. Roy. “Power spectra and spatial pattern dynamics of a ring laser,." J. Stat. Phys., 54, 1223 (1989).
251. Zauderer, E. Partial Differential Equations of Applied Mathematics. Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts, 3rd edition. Wiley, New York, 2006.
252. Ziegler, H. Principles of Structural Stability. Blaisdell, New York, 1968.
253. Zwanzig, R. “Nonlinear generalized Langevin equations." J. Stat. Phys., 9, 215–220 (1973).
254. Życzkowski, M. and A. Gajewski. “Optimal structural design in non conservative problems of elastic stability." Instability of Continuous Systems, H. Leipholz, Ed. IUTAM Symposium Herrnalb, Springer-Verlag, New York, 295–301 (1971).