Stochastic Systems

  • Christian Kuehn
Part of the Applied Mathematical Sciences book series (AMS, volume 191)


The interaction between noise and multiscale dynamics is already a large area, and it is still a field of intensive research. This chapter aims to provide a number of diverse and interlinked techniques that reflect some recent developments.


  1. [ABK12]
    D.C. Antonopoulou, D. Blömker, and G.D. Karali. Front motion in the one-dimensional stochastic Cahn–Hilliard equation. SIAM J. Math. Anal., 44(5):3242–3280, 2012.zbMATHMathSciNetGoogle Scholar
  2. [Abr10]
    R.V. Abramov. Approximate linear response for slow variables of dynamics with explicit time scale separation. J. Comp. Phys., 229(20):7739–7746, 2010.zbMATHMathSciNetGoogle Scholar
  3. [AF92]
    M. Abbad and J.A. Filar. Perturbation and stability theory for Markov control problems. IEEE Trans. Aut. Contr., 37(9): 1415–1420, 1992.zbMATHMathSciNetGoogle Scholar
  4. [AF95]
    M. Abbad and J.A. Filar. Algorithms for singularly perturbed Markov control problems: a survey. Contr. Dynamic Syst., 73:257–287, 1995.Google Scholar
  5. [AF03]
    R.A. Adams and J.J.F. Fournier. Sobolev Spaces. Elsevier, 2003.Google Scholar
  6. [AFB90]
    M. Abbad, J.A. Filar, and T.R. Bielecki. Algorithms for singularly perturbed limiting average Markov control problems. Decision and Control: Proc. 29th IEEE Conf., pages 1402–1497, 1990.Google Scholar
  7. [AFH13]
    K.E. Avrachenkov, J.A. Filar, and P.G. Howett. Analytic Perturbation Theory and Its Applications. SIAM, 2013.Google Scholar
  8. [AG13]
    G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear.Google Scholar
  9. [AK13a]
    R.V. Abramov and M.P. Kjerland. The response of reduced models of multiscale dynamics to small external perturbations. arXiv:1305.0862, pages 1–20, 2013.Google Scholar
  10. [Arn74]
    L. Arnold. Stochastic Differential Equations: Theory and Applications. Wiley, 1974.Google Scholar
  11. [Arn95]
    L. Arnold. Random dynamical systems. In Dynamical Systems (Montecatini Terme, 1994), pages 1–43. Springer, 1995.Google Scholar
  12. [Arn01]
    L. Arnold. Recent progress in stochastic bifurcation theory. In IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics, pages 15–27. Springer, 2001.Google Scholar
  13. [Arn03]
    L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg, Germany, 2003.Google Scholar
  14. [Arr89]
    S. Arrhenius. Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren. Zeitschr. Phys. Chem., 4:226–248, 1889.Google Scholar
  15. [Art99b]
    Z. Artstein. Singularly perturbed ordinary differential equations with nonautonomous fast dynamics. J. Dyn. Diff. Eq., 11(2):297–318, 1999.zbMATHMathSciNetGoogle Scholar
  16. [Bak96]
    V.I. Bakhtin. Averaging along a Markov chain. Funct. Anal. Appl., 30(1):42–44, 1996.zbMATHMathSciNetGoogle Scholar
  17. [Bak00]
    V.I. Bakhtin. Cramér asymptotics in a system with slow and fast Markovian motions. Theor. Prob. Appl., 44(1):1–17, 2000.MathSciNetGoogle Scholar
  18. [Bak03]
    V.I. Bakhtin. Cramér’s asymptotics in systems with fast and slow motions. Stochastics Stoch. Rep., 75(5):319–341, 2003.zbMATHMathSciNetGoogle Scholar
  19. [BC94b]
    C. Berzuini and D. Clayton. Bayesian analysis of survival on multiple time scales. Statistics in Medicine, 13(8):823–838, 1994.Google Scholar
  20. [BdHN06]
    A. Bovier, F. den Hollander, and F.R. Nardi. Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary. Prob. Theory Rel. Fields, 135(2):265–310, 2006.zbMATHGoogle Scholar
  21. [BdHS10]
    A. Bovier, F. den Hollander, and C. Spitoni. Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures. Ann. Prob., 38(2):661–713, 2010.zbMATHGoogle Scholar
  22. [BEGK04]
    A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein. Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Euro. Math. Soc., 6(4):399–424, 2004.Google Scholar
  23. [Bel60]
    R. Bellman. Introduction to Matrix Analysis. McGraw-Hill, 1960.Google Scholar
  24. [Ber13]
    N. Berglund. Kramers’ law: validity, derivations and generalisations. Markov Processes Relat. Fields, pages 1–24, 2013. to appear.Google Scholar
  25. [BF91]
    T.R. Bielecki and J.A. Filar. Singularly perturbed Markov control problem: limiting average cost. Annals of Operations Research, 28(1):152–168, 1991.MathSciNetGoogle Scholar
  26. [BFF99]
    D. Brown, J. Feng, and S. Feerick. Variability of firing of Hodgkin–Huxley and FitzHugh–Nagumo neurons with stochastic synaptic input. Phys. Rev. Lett., 82(23):4731–4734, 1999.Google Scholar
  27. [BG02a]
    N. Berglund and B. Gentz. As sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential. Ann. Appl. Prob., 12(4):1419–1470, 2002.zbMATHMathSciNetGoogle Scholar
  28. [BG02b]
    N. Berglund and B. Gentz. Beyond the Fokker–Planck equation: pathwise control of noisy bistable systems. J. Phys. A: Math. Gen., 35:2057–2091, 2002.zbMATHMathSciNetGoogle Scholar
  29. [BG02c]
    N. Berglund and B. Gentz. The effect of additive noise on dynamical hysteresis. Nonlinearity, 15: 605–632, 2002.zbMATHMathSciNetGoogle Scholar
  30. [BG02d]
    N. Berglund and B. Gentz. Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations. Stoch. Dyn., 2:327–356, 2002.zbMATHMathSciNetGoogle Scholar
  31. [BG02e]
    N. Berglund and B. Gentz. Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theory Related Fields, 3:341–388, 2002.MathSciNetGoogle Scholar
  32. [BG03]
    N. Berglund and B. Gentz. Geometric singular perturbation theory for stochastic differential equations. J. Diff. Eqs., 191:1–54, 2003.zbMATHMathSciNetGoogle Scholar
  33. [BG04]
    N. Berglund and B. Gentz. On the noise-induced passage through an unstable periodic orbit I: Two-level model. J. Statist. Phys., 114(5):1577–1618, 2004.zbMATHMathSciNetGoogle Scholar
  34. [BG06]
    N. Berglund and B. Gentz. Noise-Induced Phenomena in Slow–Fast Dynamical Systems. Springer, 2006.Google Scholar
  35. [BG07a]
    V.S. Borkar and V. Gaitsgory. Averaging of singularly perturbed controlled stochastic differential equations. Appl. Math. Optim., 56(2):169–209, 2007.zbMATHMathSciNetGoogle Scholar
  36. [BG09]
    N. Berglund and B. Gentz. Stochastic dynamic bifurcations and excitability. In C. Laing and G. Lord, editors, Stochastic methods in Neuroscience, volume 2, pages 65–93. OUP, 2009.Google Scholar
  37. [BG13a]
    N. Berglund and B. Gentz. On the noise-induced passage through an unstable periodic orbit II: The general case. SIAM J. Math. Anal., 2013. accepted, to appear.Google Scholar
  38. [BG13b]
    N. Berglund and B. Gentz. Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond. Electronic J. Probability, 18(24):1–58, 2013.MathSciNetGoogle Scholar
  39. [BGK05]
    A. Bovier, V. Gayrard, and M. Klein. Metastability in reversible diffusion processes. II. Precise estimates for small eigenvalues. J. Euro. Math. Soc., 7:69–99, 2005.Google Scholar
  40. [BGK12]
    N. Berglund, B. Gentz, and C. Kuehn. Hunting French ducks in a noisy environment. J. Differential Equat., 252(9):4786–4841, 2012.zbMATHMathSciNetGoogle Scholar
  41. [BGK13]
    N. Berglund, B. Gentz, and C. Kuehn. From random Poincaré maps to stochastic mixed-mode-oscillation patterns. arXiv:1312. 6353, pages 1–55, 2013.Google Scholar
  42. [BGW10]
    D. Blömker, B. Gawron, and T. Wanner. Nucleation in the one-dimensional stochastic Cahn–Hilliard model. Discr. Cont. Dyn. Syst. A, 27:25–52, 2010.zbMATHGoogle Scholar
  43. [BH04]
    D. Blömker and M. Hairer. Multiscale expansion of invariant measures for SPDEs. Comm. Math. Phys., 251(3):515–555, 2004.zbMATHMathSciNetGoogle Scholar
  44. [BHJ94]
    R. Bartussek, P. Hänggi, and P. Jung. Stochastic resonance in optical bistable systems. Phys. Rev. E, 49(5):3930, 1994.Google Scholar
  45. [BHP05]
    D. Blömker, M. Hairer, and G.A. Pavliotis. Modulation equation for SPDEs on large domains. Comm. Math. Phys., 258:479–512, 2005.zbMATHMathSciNetGoogle Scholar
  46. [BHP07]
    D. Blömker, M. Hairer, and G.A. Pavliotis. Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities. Nonlinearity, 20(7):1721–1744, 2007.zbMATHMathSciNetGoogle Scholar
  47. [BJBMS82]
    E. Ben-Jacob, D.J. Bergman, B.J. Matkowsky, and Z. Schuss. Lifetime of oscillatory steady states. Phys. Rev. A, 26(5):2805, 1982.Google Scholar
  48. [BK04]
    V.I. Bakhtin and Yu. Kifer. Diffusion approximation for slow motion in fully coupled averaging. Probab. Theory Relat. Fields, 129:157–181, 2004.zbMATHMathSciNetGoogle Scholar
  49. [BKLC10]
    P. Borowski, R. Kuske, X.-X. Li, and J.L. Cabrera. Characterizing mixed mode oscillations shaped by noise and bifurcation structure. Chaos, 20:043117, 2010.MathSciNetGoogle Scholar
  50. [BKM12]
    L.L. Bonilla, A. Klar, and S. Martin. Higher order averaging of linear Fokker–Planck equations with periodic forcing. SIAM J. Appl. Math., 72(4):1315–1342, 2012.zbMATHMathSciNetGoogle Scholar
  51. [BKPR06]
    K. Ball, T.G. Kurtz, L. Popovic, and G. Rempala. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Prob., 16(4):1925–1961, 2006.zbMATHMathSciNetGoogle Scholar
  52. [BKWL12]
    P. Bokes, J.R. King, A.T.A. Wood, and M. Loose. Multiscale stochastic modelling of gene expression. J. Math. Biol., 65:493–520, 2012.zbMATHMathSciNetGoogle Scholar
  53. [BL12]
    N. Berglund and D. Landon. Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh–Nagumo model. Nonlinearity, 25:2303–2335, 2012.zbMATHMathSciNetGoogle Scholar
  54. [Blö03]
    D. Blömker. Amplitude equations for locally cubic nonautonomous nonlinearities. SIAM J. Appl. Dyn. Syst., 2(3):464–486, 2003.zbMATHMathSciNetGoogle Scholar
  55. [Blö07]
    D. Blömker. Amplitude Equations for Stochastic Partial Differential Equations. World Scientific, 2007.Google Scholar
  56. [BLP12]
    C. Le Bris, T. Lelièvre, and M. Perez. A mathematical formalization of the parallel replica dynamics. Monte Carlo Meth. Appl., 18(2):119–146, 2012.zbMATHGoogle Scholar
  57. [BM09]
    D. Blömker and W.W. Mohammed. Amplitude equation for SPDEs with quadratic nonlinearities. Electron. J. Prob., 14(88):2527–2550, 2009.zbMATHGoogle Scholar
  58. [BM11]
    A.J. Black and A.J. McKane. WKB calculation of an epidemic outbreak distribution. J. Stat. Mech., 2011:P12006, 2011.Google Scholar
  59. [BN13]
    P.C. Bressloff and J.M. Newby. Metastability in a stochastic neural network modeled as a velocity jump Markov process. SIAM J. Appl. Dyn Syst., 12:1394–1435, 2013.zbMATHMathSciNetGoogle Scholar
  60. [BO99]
    C.M. Bender and S.A. Orszag. Asymptotic Methods and Perturbation Theory. Springer, 1999.Google Scholar
  61. [Bob07]
    R.V. Bobryk. Closure method and asymptotic expansions for linear stochastic systems. J. Math. Anal. Appl., 329(1):703–711, 2007.zbMATHMathSciNetGoogle Scholar
  62. [Bor97]
    V.S. Borkar. Stochastic approximation with two time scales. Syst. Contr. Lett., 29(5):291–294, 1997.zbMATHMathSciNetGoogle Scholar
  63. [BPH94]
    L. Bocquet and J. Piasecki and J.-P. Hansen. On the Brownian motion of a massive sphere suspended in a hard-sphere fluid. I. Multiple-time-scale analysis and microscopic expression for the friction coefficient. J. Statist. Phys., 76:505–526, 1994.Google Scholar
  64. [BPSV82]
    R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani. Stochastic resonance in climatic change. Tellus, 34(11):10–16, 1982.Google Scholar
  65. [BPSV83]
    R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani. A theory of stochastic resonance in climatic change. SIAM J. Appl. Math., 43(3):565–578, 1983.MathSciNetGoogle Scholar
  66. [Bré12a]
    C.-E. Bréhier. Strong and weak orders in averaging for SPDEs. Stoch. Proc. Appl., 122(7):2553–2593, 2012.zbMATHGoogle Scholar
  67. [Bro28]
    R. Brown. A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag., 4:161–173, 1828.Google Scholar
  68. [BS79]
    G. Blankenship and S. Sachs. Singularly perturbed linear stochastic ordinary differential equations. SIAM J. Math. Anal., 1979:306–320, 1979.MathSciNetGoogle Scholar
  69. [BS82]
    B.Z. Bobrovsky 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.zbMATHMathSciNetGoogle Scholar
  70. [BSV81]
    R. Benzi, A. Sutera, and A. Vulpiani. The mechanism of stochastic resonance. J. Phys. A, 14(11): 453–457, 1981.MathSciNetGoogle Scholar
  71. [BY02]
    G. Badowski and G.G. Yin. Stability of hybrid dynamic systems containing singularly perturbed random processes. IEEE Trans. Aut. Contr., 47(12):2021–2032, 2002.MathSciNetGoogle Scholar
  72. [CD09]
    N. Chernov and D. Dolgopyat. Brownian Brownian motion, volume 927 of Mem. Amer. Math. Soc. AMS, 2009.Google Scholar
  73. [CDG+98]
    J.C. Celet, D. Dangoisse, P. Glorieux, G. Lythe, and T. Erneux. Slowly passing through resonance strongly depends on noise. Phys. Rev. Lett., 81(5):975–978, 1998.Google Scholar
  74. [Cer09]
    S. Cerrai. A Khasminskii type averaging principle for stochastic reaction–diffusion equations. Ann. Appl. Prob., 19(3):899–948, 2009.zbMATHMathSciNetGoogle Scholar
  75. [Cer11]
    S. Cerrai. Averaging principle for systems of reaction–diffusion equations with polynomial nonlinearities perturbed by multiplicative noise. SIAM J. Math. Anal., 43(6):2482–2518, 2011.zbMATHMathSciNetGoogle Scholar
  76. [CF94]
    H. Crauel and F. Flandoli. Attractors for random dynamical systems. Probab. Theory Relat. Fields, 100(3):365–393, 1994.zbMATHMathSciNetGoogle Scholar
  77. [CF06a]
    S. Cerrai and M. Freidlin. On the Smoluchowski–Kramers approximation for a system with an infinite number of degrees of freedom. Probab. Theory Rel. Fields, 135(3):363–394, 2006.zbMATHMathSciNetGoogle Scholar
  78. [CF06b]
    S. Cerrai and M. Freidlin. Smoluchowski–Kramers approximation for a general class of SPDEs. J. Evol. Equat., 6(4):657–689, 2006.zbMATHMathSciNetGoogle Scholar
  79. [CF09]
    S. Cerrai and M. Freidlin. Averaging principle for a class of stochastic reaction–diffusion equations. Probab. Theory Rel. Fields, 144(1):137–177, 2009.zbMATHMathSciNetGoogle Scholar
  80. [CFM12]
    P. Chleboun, A. Faggionato, and F. Martinelli. Time scale separation and dynamic heterogeneity in the low temperature east model. arXiv:1212.2399v1, pages 1–40, 2012.Google Scholar
  81. [CFNS09]
    P. Channell, I. Fuwape, A.B. Neiman, and A. Shilnikov. Variability of bursting patterns in a neuron model in the presence of noise. J. Comp. Neurosci., 27(3):527–542, 2009.MathSciNetGoogle Scholar
  82. [CGOV84]
    M. Cassandro, A. Galves, E. Olivieri, and M.E. Vares. Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys., 35(5):603–634, 1984.zbMATHMathSciNetGoogle Scholar
  83. [Che82]
    N.N. Chentsova. An investigation of a certain model system of quasi-stochastic relaxation oscillations. Russ. Math. Surv., 37(5):164, 1982.Google Scholar
  84. [CK88]
    T.R. Chay and H.S. Kang. Role of single-channel stochastic noise on bursting clusters of pancreatic beta-cells. Biophys. J., 54(3):427–435, 1988.Google Scholar
  85. [CL11]
    C. Chipot and T. Lelièvre. Enhanced sampling of multidimensional free-energy landscapes using adaptive biasing forces. SIAM J. Appl. Math., 71(5):1673–1695, 2011.zbMATHMathSciNetGoogle Scholar
  86. [CMR13]
    G.W.A. Constable, A.J. McKane, and T. Rogers. Stochastic dynamics on slow manifolds. J. Phys. A, 46:295002, 2013.MathSciNetGoogle Scholar
  87. [CR03]
    S. Cerrai and M. Röckner. Large deviations for invariant measures of general stochastic reaction–diffusion systems. Comptes Rendus Acad. Sci. Paris S. I Math., 337:597–602, 2003.zbMATHGoogle Scholar
  88. [CR04]
    S. Cerrai and M. Röckner. Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Annals of Probability, 32:1–40, 2004.MathSciNetGoogle Scholar
  89. [CR05]
    S. Cerrai and M. Röckner. Large deviations for invariant measures of stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Annales de l’Institut Henri Poincaré (B), 41:69–105, 2005.Google Scholar
  90. [CSRR08]
    R. Curtu, A. Shpiro, N. Rubin, and J. Rinzel. Mechanisms for frequency control in neuronal competition models. SIAM J. Appl. Dyn. Syst., 7(2):609–649, 2008.zbMATHMathSciNetGoogle Scholar
  91. [CWS83b]
    M. Coderch, A.S. Willsky, and S.S. Sastry. Hierarchical aggregation of singularly perturbed finite state Markov processes. Stochastics, 8(4):259–289, 1983.zbMATHMathSciNetGoogle Scholar
  92. [CWS10]
    A.F: Cheviakov, M.J. Ward, and R. Straube. An asymptotic analysis of the mean first passage time for narrow escape problems: Part II: The sphere. Multiscale Model. Simul., 8(3):836–870, 2010.Google Scholar
  93. [Day83]
    M.V. Day. On the exponential exit law in the small parameter exit problem. Stochastics, 8(4):297–323, 1983.zbMATHMathSciNetGoogle Scholar
  94. [Day84]
    M.V. Day. On the asymptotic relation between equilibrium density and exit measure in the exit problem. Stochastics, 12(3):303–330, 1984.zbMATHMathSciNetGoogle Scholar
  95. [Day87]
    M.V. Day. Recent progress on the small parameter exit problem. Stochastics, 20(2):121–150, 1987.zbMATHMathSciNetGoogle Scholar
  96. [Day89]
    M.V. Day. Boundary local time and small parameter exit problems with characteristic boundaries. SIAM J. Math. Anal., 20(1):222–248, 1989.zbMATHMathSciNetGoogle Scholar
  97. [Day90]
    M.V. Day. Large deviations results for the exit problem with characteristic boundary. J. Math. Anal. Appl., 147(1):134–153, 1990.zbMATHMathSciNetGoogle Scholar
  98. [Day92]
    M.V. Day. Conditional exits for small noise diffusions with characteristic boundary. Ann. Prob., 20(3):1385–1419, 1992.zbMATHGoogle Scholar
  99. [Day94]
    M.V. Day. Cycling and skewing of exit measures for planar systems. Stochastics, 48(3):227–247, 1994.zbMATHMathSciNetGoogle Scholar
  100. [Day95]
    M.V. Day. On the exit law from saddle points. Stoch. Proc. Appl., 60(2):287–311, 1995.zbMATHGoogle Scholar
  101. [dBM87]
    C. Van den Broeck and P. Mandel. Delayed bifurcations in the presence of noise. Phys. Lett. A, 122: 36–38, 1987.Google Scholar
  102. [DD07]
    A. Du and J. Duan. Invariant manifold reduction for stochastic dynamical systems. Dyn. Syst. Appl., 16:681–696, 2007.zbMATHMathSciNetGoogle Scholar
  103. [DEF74]
    A. Devinatz, R. Ellis, and A. Friedman. The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II. Ind. Univ. Math. J., 23(11):991–1011, 1974.Google Scholar
  104. [Del83]
    F. Delebecque. A reduction process for perturbed Markov chains. SIAM J. Appl. Math., 43(2):325–350, 1983.zbMATHMathSciNetGoogle Scholar
  105. [DG93a]
    D. Dawson and A. Greven. Hierarchical models of interacting diffusions: multiple time scale phenomena, phase transition and pattern of cluster-formation. Probab. Theory Related Fields, 96(4):435–473, 1993.zbMATHMathSciNetGoogle Scholar
  106. [DG93b]
    D. Dawson and A. Greven. Multiple time scale analysis of interacting diffusions. Probab. Theory Related Fields, 95(4):467–508, 1993.zbMATHMathSciNetGoogle Scholar
  107. [dH04]
    F. den Hollander. Metastability under stochastic dynamics. Stoch. Proc. Appl., 114:1–26, 2004.zbMATHGoogle Scholar
  108. [dH08]
    F. den Hollander. Large Deviations. Amer. Math. Soc., 2008.Google Scholar
  109. [DLS03]
    J. Duan, K. Lu, and B. Schmalfuss. Invariant manifolds for stochastic partial differential equations. Ann. Prob., 31(4):2109–2135, 2003.zbMATHMathSciNetGoogle Scholar
  110. [DLS04]
    J. Duan, K. Lu, and B. Schmalfuss. Smooth stable and unstable manifolds for stochastic evolutionary equations. J. Dyn. Diff. Eq., 16(4):949–972, 2004.zbMATHMathSciNetGoogle Scholar
  111. [DMS86a]
    M.M. Dygas, 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.zbMATHMathSciNetGoogle Scholar
  112. [DMS86b]
    M.M. Dygas, B.J. Matkowsky, and Z. Schuss. A singular perturbation approach to non-Markovian escape rate problems with state dependent friction. J. Chem. Phys., 84:3731, 1986.Google Scholar
  113. [DP96]
    J.D. Deuschel and A. Pisztora. Surface order large deviations for high-density percolation. Probab. Theory Rel. Fields, 104(4):467–482, 1996.zbMATHMathSciNetGoogle Scholar
  114. [DRL12]
    A. Destexhe and M. Rudolph-Lilith. Neuronal Noise. Springer, 2012.Google Scholar
  115. [DS89a]
    J.D. Deuschel and D.W. Stroock. Large Deviations. Academic Press, 1989.Google Scholar
  116. [DSW12]
    P. Dupuis, K. Spiliopoulos, and H. Wang. Importance sampling for multiscale diffusions. Multiscale Model. Simul., 10(1):1–27, 2012.zbMATHMathSciNetGoogle Scholar
  117. [Dur85]
    J. Durbin. The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob., 22:99–122, 1985.zbMATHMathSciNetGoogle Scholar
  118. [Dur92]
    J. Durbin. The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob., 29:291–304, 1992. with an appendix by D. Williams.Google Scholar
  119. [Dur94]
    R. Durrett. The Essentials of Probability. Duxbury, 1994.Google Scholar
  120. [Dur10a]
    R. Durrett. Probability: Theory and Examples - 4th edition. CUP, 2010.Google Scholar
  121. [DV75a]
    M.D. Donsker and S.R.S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, I. Comm. Pure Appl. Math., 28(1):1–47, 1975.zbMATHMathSciNetGoogle Scholar
  122. [DV75b]
    M.D. Donsker and S.R.S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, II. Comm. Pure Appl. Math., 28(2):279–301, 1975.zbMATHGoogle Scholar
  123. [DV76]
    M.D. Donsker and S.R.S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, III. Comm. Pure Appl. Math., 29(4):389–461, 1976.zbMATHMathSciNetGoogle Scholar
  124. [DV83a]
    M.D. Donsker and S.R.S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, IV. Comm. Pure Appl. Math., 36(2):183–212, 1983.zbMATHMathSciNetGoogle Scholar
  125. [DV89]
    M.D. Donsker and S.R.S. Varadhan. Large deviations from a hydrodynamic scaling limit. Comm. Pure Appl. Math., 42(3):243–270, 1989.zbMATHMathSciNetGoogle Scholar
  126. [DVE06]
    L. DeVille and E. Vanden-Eijnden. A nontrivial scaling limit for multiscale Markov chains. J. Stat. Phys., 126(1):75–94, 2006.MathSciNetGoogle Scholar
  127. [DVEM05]
    L. DeVille, E. Vanden-Eijnden, and C.B. Muratov. Two distinct mechanisms of coherence in randomly perturbed dynamical systems. Phys. Rev. E, 72(3):031105, 2005.Google Scholar
  128. [Dys62]
    F.J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Physics, 3(6):1191, 1962.Google Scholar
  129. [DZ98]
    A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, volume 38 of Applications of Mathematics. Springer-Verlag, 1998.Google Scholar
  130. [EF90]
    A. Eizenberg and M. Freidlin. On the Dirichlet problem for a class of second order PDE systems with small parameter. Stoch. Stoch. Rep., 33(3):111–148, 1990.zbMATHMathSciNetGoogle Scholar
  131. [EF93]
    A. Eizenberg and M. Freidlin. Large deviations for Markov processes corresponding to PDE systems. Ann. Probab., 21(2): 1015–1044, 1993.zbMATHMathSciNetGoogle Scholar
  132. [Ein05]
    A. Einstein. Über die von der molekular-kinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys., pages 549–560, 1905.Google Scholar
  133. [ERVE02]
    W. E., W. Ren, and E. Vanden-Eijnden. String method for the study of rare events. Phys. Rev. B, 66(5):052301, 2002.Google Scholar
  134. [ERVE05]
    W. E., W. Ren, and E. Vanden-Eijnden. Finite temperature string method for the study of rare events. J. Phys. Chem. B, 109(14):6688–6693, 2005.Google Scholar
  135. [ESY11]
    L. Erdös, B. Schlein, and H.T. Yau. Universality of random matrices and local relaxation flow. Invent. Math., 185(1):75–119, 2011.zbMATHMathSciNetGoogle Scholar
  136. [Eva02]
    L.C. Evans. Partial Differential Equations. AMS, 2002.Google Scholar
  137. [EVE04]
    W. E and E. Vanden-Eijnden. Metastability, conformation dynamics, and transition pathways in complex systems. In Multiscale Modelling and Simulation, pages 35–68. Springer, 2004.Google Scholar
  138. [EVE06]
    W. E and E. Vanden-Eijnden. Towards a theory of transition paths. J. Stat. Phys., 123(3):503–523, 2006.Google Scholar
  139. [EVE10]
    W. E and E. Vanden-Eijnden. Transition-path theory and path-finding algorithms for the study of rare events. Phys. Chem., 61:391–420, 2010.Google Scholar
  140. [Eyr35]
    H. Eyring. The activated complex in chemical reactions. J. Chem. Phys., 3:107–115, 1935.Google Scholar
  141. [FD11]
    H. Fu and J. Duan. An averaging principle for two time-scale stochastic partial differential equations. Stoch. Dyn., 11:353–367, 2011.zbMATHMathSciNetGoogle Scholar
  142. [FG13]
    J.E. Frank and G.A. Gottwald. Stochastic homogenization for an energy conserving multi-scale toy model of the atmosphere. Physica D, 254:46–56, 2013.MathSciNetGoogle Scholar
  143. [FJ92]
    W.H. Fleming and M.R. James. Asymptotic series and exit time probabilities. Ann. Probab., 20(3):1369–1384, 1992.zbMATHMathSciNetGoogle Scholar
  144. [FLD13]
    H. Fu, X. Liu, and J. Duan. Slow manifolds for multi-time-scale stochastic evolutionary systems. Comm. Math. Sci., 11(1):141–162, 2013.zbMATHMathSciNetGoogle Scholar
  145. [Fle71]
    W.H. Fleming. Stochastic control for small noise intensities. SIAM J. Control, 9(3):473–517, 1971.MathSciNetGoogle Scholar
  146. [Fle77]
    W.H. Fleming. Exit probabilities and optimal stochastic control. Appl. Math. Optim., 4(1):329–346, 1977.MathSciNetGoogle Scholar
  147. [FMVE05]
    C. Franzke, A.J. Majda, and E. Vanden-Eijnden. Low-order stochastic mode reduction for a realistic barotropic model climate. J. Atmos. Sci., 62:1722–1745, 2005.MathSciNetGoogle Scholar
  148. [Fol99]
    G. Folland. Real Analysis - Modern Techniques and Their Applications. Wiley, 1999.Google Scholar
  149. [Fox97]
    R.F. Fox. Stochastic versions of the Hodgkin–Huxley equations. Biophys. J., 72:2068–2074, 1997.Google Scholar
  150. [Fre73]
    M.I. Freidlin. The action functional for a class of stochastic processes. Theory Probab. Appl., 17(3): 511–515, 1973.Google Scholar
  151. [Fre78]
    M.I. Freidlin. The averaging principle and theorems on large deviations. Russ. Math. Surv., 33(5): 117–176, 1978.MathSciNetGoogle Scholar
  152. [Fre85]
    M.I. Freidlin. Limit theorems for large deviations and reaction–diffusion equations. Ann. Probab., 13(3):639–675, 1985.zbMATHMathSciNetGoogle Scholar
  153. [Fre91]
    M.I. Freidlin. Coupled reaction–diffusion equations. Ann. Probab., 19(1):29–57, 1991.zbMATHMathSciNetGoogle Scholar
  154. [Fre96]
    M.I. Freidlin. Markov Processes and Differential Equations: Asymptotic Problems. Springer, 1996.Google Scholar
  155. [Fre00a]
    M.I. Freidlin. Diffusion processes on graphs: stochastic differential equations, large deviation principle. Probab. Theory Rel. Fields, 116(2):181–220, 2000.zbMATHMathSciNetGoogle Scholar
  156. [Fre00b]
    M.I. Freidlin. Quasi-deterministic approximation, metastability and stochastic resonance. Physica D, 137(3):333–352, 2000.zbMATHMathSciNetGoogle Scholar
  157. [Fre01]
    M.I. Freidlin. On stable oscillations and equilibriums induced by small noise. J. Stat. Phys., 103(1):283–300, 2001.zbMATHMathSciNetGoogle Scholar
  158. [Fre04]
    M.I. Freidlin. Some remarks on the Smoluchowski–Kramers approximation. J. Stat. Phys., 117(3): 617–634, 2004.zbMATHMathSciNetGoogle Scholar
  159. [Fri06b]
    A. Friedman. Stochastic Differential Equations and Applications. Dover, 2006.Google Scholar
  160. [FS99a]
    M.I. Freidlin and R.B. Sowers. A comparison of homogenization and large deviations, with applications to wavefront propagation. Stoch. Proc. Appl., 82(1):23–52, 1999.zbMATHMathSciNetGoogle Scholar
  161. [FW92]
    M.I. Freidlin and A.D. Wentzell. Reaction-diffusion equations with randomly perturbed boundary conditions. Ann. Probab., 20(2):963–986, 1992.zbMATHMathSciNetGoogle Scholar
  162. [FW93]
    M.I. Freidlin and A.D. Wentzell. Diffusion processes on graphs and the averaging principle. Ann. Probab., 21(4):2215–2245, 1993.zbMATHMathSciNetGoogle Scholar
  163. [FW98a]
    M.I. Freidlin and M. Weber. Random perturbations of nonlinear oscillators. Ann. Probab., 26(3): 925–967, 1998.zbMATHMathSciNetGoogle Scholar
  164. [FW98b]
    M.I. Freidlin and A.D. Wentzell. Random Perturbations of Dynamical Systems. Springer, 1998.Google Scholar
  165. [FW99]
    M.I. Freidlin and M. Weber. A remark on random perturbations of the nonlinear pendulum. Ann. Appl. Probab., 9(3):611–628, 1999.zbMATHMathSciNetGoogle Scholar
  166. [FW04]
    M.I. Freidlin and A.D. Wentzell. Random perturbations of dynamical systems and diffusion processes with conservation laws. Probab. Theory Rel. Fields, 128(3):441–466, 2004.zbMATHMathSciNetGoogle Scholar
  167. [FW06]
    M.I. Freidlin and A.D. Wentzell. Long-time behavior of weakly coupled oscillators. J. Stat. Phys., 123(6):1311–1337, 2006.zbMATHMathSciNetGoogle Scholar
  168. [Gam95]
    L. Gammaitoni. Stochastic resonance and the dithering effect in threshold physical systems. Phys. Rev. E, 52(5):4691, 1995.Google Scholar
  169. [Gar09]
    C. Gardiner. Stochastic Methods. Springer, Berlin Heidelberg, Germany, 4th edition, 2009.zbMATHGoogle Scholar
  170. [Gem79]
    S. Geman. Some averaging and stability results for random differential equations. SIAM J. Appl. Math., 36(1):86–105, 1979.zbMATHMathSciNetGoogle Scholar
  171. [GH99]
    J. Grasman and O.A. Van Herwaarden. Asymptotic Methods for the Fokker–Planck Equation and the Exit Problem in Applications. Springer, 1999.Google Scholar
  172. [GH00]
    I. Goychuk and P. Hänggi. Stochastic resonance in ion channels characterized by information theory. Phys. Rev. E, 61(4):4272, 2000.Google Scholar
  173. [GHJM98]
    L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni. Stochastic resonance. Rev. Mod. Phys., 70: 223–287, 1998.Google Scholar
  174. [GK04]
    D. Givon and R. Kupferman. White noise limits for discrete dynamical systems driven by fast deterministic dynamics. Physica A, 335(3):385–412, 2004.MathSciNetGoogle Scholar
  175. [GKT97]
    M. Grossglauser, S. Keshav, and D.N. Tse. RCBR: a simple and efficient service for multiple time-scale traffic. IEEE/ACM Trans. Netw., 5(6):741–755, 1997.Google Scholar
  176. [GL05]
    A. Guillin and R. Liptser. MDP for integral functionals of fast and slow processes with averaging. Stochastic Process. Appl., 115(7):1187–1207, 2005.zbMATHMathSciNetGoogle Scholar
  177. [GMG07]
    G. Gigante, M. Mattia, and P.D. Giudice. Diverse population-bursting modes of adapting spiking neurons. Phys. Rev. Lett., 98(14):148101, 2007.Google Scholar
  178. [GMGW98]
    T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller. Random-matrix theories in quantum physics: common concepts. Phys. Rep., 299(4):189–425, 1998.MathSciNetGoogle Scholar
  179. [GMMSS89]
    L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta, and S. Santucci. Stochastic resonance in bistable systems. Phys. Rev. Lett., 62(4):349–352, 1989.Google Scholar
  180. [GMS95]
    L. Gammaitoni, F. Marchesoni, and S. Santucci. Stochastic resonance as a bona fide resonance. Phys. Rev. Lett., 74(7):1052–1055, 1995.Google Scholar
  181. [GMSS+89]
    L. Gammaitoni, E. Menichella-Saetta, S. Santucci, F. Marchesoni, and C. Presilla. Periodically time-modulated bistable systems: stochastic resonance. Phys. Rev. A, 40(4):2114, 1989.Google Scholar
  182. [GOV87a]
    A. Galves, E. Olivieri, and M.E. Vares. Metastability for a class of dynamical systems subject to small random perturbations. Ann. Prob., 15(4):1288–1305, 1987.zbMATHMathSciNetGoogle Scholar
  183. [GT12a]
    A. Genadot and M. Thieullen. Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions. Adv. Appl. Probab., 44(3):749–773, 2012.zbMATHMathSciNetGoogle Scholar
  184. [GT12b]
    A. Genadot and M. Thieullen. Multiscale piecewise deterministic Markov process in infnite dimension: central limit theorem and Langevin approximation. arXiv:1211.1894v1, pages 1–33, 2012.Google Scholar
  185. [Gup12]
    A. Gupta. The Fleming–Viot limit of an interacting spatial population with fast density regulation. Electronic J. Probab., 17(104):1–55, 2012.Google Scholar
  186. [GX01]
    P.-L. Gong and J.-X. Xu. Global dynamics and stochastic resonance of the forced FitzHugh–Nagumo neuron model. Phys. Rev. E, 63(3):031906, 2001.Google Scholar
  187. [Hän02]
    P. Hänggi. Stochastic resonance in biology how noise can enhance detection of weak signals and help improve biological information processing. ChemPhysChem, 3(3):285–290, 2002.Google Scholar
  188. [HBB13a]
    M. Hasler, V. Belykh, and I. Belykh. Dynamics of stochastically blinking systems. Part I: Finite time properties. SIAM J. Appl. Dyn. Syst., 12(2):1007–1030, 2013.Google Scholar
  189. [HBB13b]
    M. Hasler, V. Belykh, and I. Belykh. Dynamics of stochastically blinking systems. Part II: Asymptotic properties. SIAM J. Appl. Dyn. Syst., 12(2):1031–1084, 2013.Google Scholar
  190. [HDFS06]
    I. Horenko, E. Dittmer, A. Fischer, and C. Schütte. Automated model reduction for complex systems exhibiting metastability. Multiscale Model. Simul., 5(3):802–827, 2006.zbMATHMathSciNetGoogle Scholar
  191. [HE05]
    R.C. Hilborn and R.J. Erwin. Fokker–Planck analysis of stochastic coherence in models of an excitable neuron with noise in both fast and slow dynamics. Phys. Rev. E, 72:031112, 2005.MathSciNetGoogle Scholar
  192. [Hel02]
    B. Helffer. Semiclassical Analysis, Witten Laplacians and Statistical Mechanics. World Scientific, 2002.Google Scholar
  193. [HHS08]
    F. Hérau, M. Hitrik, and J. Sjöstrand. Tunnel effect for Kramers–Fokker–Planck type operators. Ann. Henri Poincaré, 9(2):209–275, 2008.zbMATHGoogle Scholar
  194. [HI02]
    S. Herrmann and P. Imkeller. Barrier crossings characterize stochastic resonance. Stoch. Dyn., 2(3):413–436, 2002.zbMATHMathSciNetGoogle Scholar
  195. [HI05]
    S. Herrmann and P. Imkeller. The exit problem for diffusions with time-periodic drift and stochastic resonance. Ann. Appl. Probab., 15:39–68, 2005.zbMATHMathSciNetGoogle Scholar
  196. [Hig01]
    D.J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 43(3):525–546, 2001.zbMATHMathSciNetGoogle Scholar
  197. [HIP08]
    S. Herrmann, P. Imkeller, and D. Peithmann. Large deviations and a Kramers type law for self-stabilizing diffusions. Ann. Appl. Probab., 18(4):1379–1423, 2008.zbMATHMathSciNetGoogle Scholar
  198. [HJZM93]
    P. Hänggi, P. Jung, C. Zerbe, and F. Moss. Can colored noise improve stochastic resonance? J. Stat. Phys., 70(1):25–47, 1993.zbMATHGoogle Scholar
  199. [HL00a]
    S. Habib and G. Lythe. Dynamics of kinks: nucleation, diffusion, and annihilation. Phys. Rev. Lett., 84(6):1070–1073, 2000.Google Scholar
  200. [HL06]
    W. Horsthemke and R. Lefever. Noise-Induced Transitions. Springer, 2006.Google Scholar
  201. [HM09c]
    P. Hitczenko and G.S. Medvedev. Bursting oscillations induced by small noise. SIAM J. Appl. Math., 69(5):1359–1392, 2009.zbMATHMathSciNetGoogle Scholar
  202. [HM10]
    M. Hairer and A.J. Majda. A simple framework to justify linear response theory. Nonlinearity, 23(4):909–922, 2010.zbMATHMathSciNetGoogle Scholar
  203. [HMS04]
    W. Huisinga, S. Meyn, and C. Schütte. Phase transitions and metastability in Markovian and molecular systems. Ann. Prob., 14(1):419–458, 2004.zbMATHGoogle Scholar
  204. [HN04]
    B. Helffer and F. Nier. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary. Mat. Contemp., 26:41–85, 2004.zbMATHMathSciNetGoogle Scholar
  205. [HN05]
    B. Helffer and F. Nier. Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, volume 1862 of Lect. Notes Math. Springer, 2005.Google Scholar
  206. [Hop95]
    F.C. Hoppenstaedt. Singular perturbation solutions of noisy systems. SIAM J. Appl. Math., 55(2): 544–551, 1995.MathSciNetGoogle Scholar
  207. [HS84]
    B. Helffer and J. Sjostrand. Multiple wells in the semi-classical limit I. Comm. Partial Diff. Equat., 9(4):337–408, 1984.zbMATHMathSciNetGoogle Scholar
  208. [HS04]
    D. Holcman and Z. Schuss. Escape through a small opening: receptor trafficking in a synaptic membrane. J. Stat. Phys., 117(5):975–1014, 2004.zbMATHMathSciNetGoogle Scholar
  209. [HTB90a]
    P. Hänggi, P. Talkner, and M. Borkovec. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys., 62(2):251–341, 1990.Google Scholar
  210. [IKY99]
    A.M. Il’in, R.Z. Khasminskii, and G. Yin. Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: rapid switchings. J. Math. Anal. Appl., 238(2):516–539, 1999.Google Scholar
  211. [IP01]
    P. Imkeller and I. Pavlyukevich. Stochastic resonance in two-state Markov chains. Archiv Math., 77:107–115, 2001.zbMATHMathSciNetGoogle Scholar
  212. [IP02]
    P. Imkeller and I. Pavlyukevich. Model reduction and stochastic resonance. Stoch. Dyn., 2(4):463–506, 2002.zbMATHMathSciNetGoogle Scholar
  213. [IP06a]
    P. Imkeller and I. Pavlyukevich. First exit times of SDEs driven by stable Lévy processes. Stoch. Process. Appl., 116(4):611–642, 2006.zbMATHMathSciNetGoogle Scholar
  214. [IP06b]
    P. Imkeller and I. Pavlyukevich. Lévy flights: transitions and meta-stability. J. Phys. A, 39(15):L237, 2006.Google Scholar
  215. [IPW09]
    P. Imkeller, I. Pavlyukevich, and T. Wetzel. First exit times for Lévy-driven diffusions with exponentially light jumps. Ann. Probab., 37(2):530–564, 2009.zbMATHMathSciNetGoogle Scholar
  216. [JH91]
    P. Jung and P. Hänggi. Amplification of small signals via stochastic resonance. Phys. Rev. A, 44(12):8032, 1991.Google Scholar
  217. [JL98]
    K.M. Jansons and G.D. Lythe. Stochastic calculus: application to dynamic bifurcations and threshold crossings. J. Stat. Phys., 90:227–251, 1998.zbMATHMathSciNetGoogle Scholar
  218. [JP04]
    J. Jacod and P. Protter. Probability Essentials. Springer, 2004.Google Scholar
  219. [JZLZ13]
    L. Ji, J. Zhang, X. Lang, and X. Zhang. Coupling and noise induced spiking-bursting transition in a parabolic bursting model. Chaos, 23:013141, 2013.Google Scholar
  220. [KB02]
    R. Kuske and S.M. Baer. Asymptotic analysis of noise sensitivity in a neuronal burster. Bull. Math. Biol., 64(3):447–481, 2002.Google Scholar
  221. [KB09a]
    R. Kuske and R. Borowski. Survival of subthreshold oscillations: the interplay of noise, bifurcation structure, and return mechanism. Discr. and Cont. Dyn. Sys. S, 2(4):873–895, 2009.zbMATHMathSciNetGoogle Scholar
  222. [KDMS88]
    M.M. Klosek-Dygas, B.J. Matkowsky, and Z. Schuss. Colored noise in dynamical systems. SIAM J. Appl. Math., 48(2):425–441, 1988.zbMATHMathSciNetGoogle Scholar
  223. [KDMS89]
    M.M. Klosek-Dygas, B.J. Matkowsky, and Z. Schuss. Colored noise in activated rate processes. J. Stat. Phys., 54(5):1309–1320, 1989.zbMATHMathSciNetGoogle Scholar
  224. [KGG07]
    R. Kuske, L.F. Gordillo, and P. Greenwood. Sustained oscilla- tions via coherence resonance in SIR. J. Theor. Biol., 245(3): 459–469, 2007.MathSciNetGoogle Scholar
  225. [KGN98]
    A.L. Kawczynski, J. Gorecki, and B. Nowakowski. Microscopic and stochastic simulations of oscillations in a simple model of chemical system. J. Phys. Chem. A, 102(36):7113–7122, 1998.Google Scholar
  226. [Kif81]
    Yu. Kifer. The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Israel J. Math., 40(1):74–96, 1981.MathSciNetGoogle Scholar
  227. [Kif90]
    Yu. Kifer. A discrete-time version of the Wentzell–Freidlin theory. Ann. Prob., 18(4):1676–1692, 1990.zbMATHMathSciNetGoogle Scholar
  228. [Kif01b]
    Yu. Kifer. Stochastic versions of Anosov’s and Neistadt’s theorems on averaging. Stoch. Dyn., 1(1): 1–21, 2001.zbMATHMathSciNetGoogle Scholar
  229. [KK01]
    R.Z. Khasminskii and N. Krylov. On averaging principle for diffusion processes with null-recurrent fast component. Stoch. Proc. Appl., 93(2):229–240, 2001.zbMATHMathSciNetGoogle Scholar
  230. [KK05]
    M.M. Klosek and R. Kuske. Multiscale analysis of stochastic delay differential equations. Multiscale Model. Simul., 3(3):706–729, 2005.zbMATHMathSciNetGoogle Scholar
  231. [KM03a]
    P. Kramer and A. Majda. Stochastic mode reduction for particle-based simulation methods for complex microfluid systems. SIAM J. Appl. Math., 64(2):401–422, 2003.zbMATHMathSciNetGoogle Scholar
  232. [KM03b]
    P. Kramer and A. Majda. Stochastic mode reduction for the immersed boundary method. SIAM J. Appl. Math., 64(2):369–400, 2003.zbMATHMathSciNetGoogle Scholar
  233. [KMS04]
    M.A. Katsoulakis, A.J. Majda, and A. Sopasakis. Multiscale couplings in prototype hybrid deterministic/stochastic systems: part I, deterministic closures. Commun. Math. Sci., 2(2): 255–294, 2004.zbMATHMathSciNetGoogle Scholar
  234. [KMZT85]
    C. Knessel, B.J. Matkowsky, Z. Schuss and C. Tier. An asymptotic theory of large deviations for Markov jump processes. SIAM J. Appl. Math., 45(6):1006–1028, 1985.MathSciNetGoogle Scholar
  235. [KMZT86a]
    C. Knessel, 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.MathSciNetGoogle Scholar
  236. [KMZT86b]
    C. Knessel, 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.MathSciNetGoogle Scholar
  237. [KMZT86c]
    C. Knessel, B.J. Matkowsky, Z. Schuss and C. Tier. A singular perturbation approach to first passage times for Markov jump processes. J. Stat. Phys., 42(1):169–184, 1986.Google Scholar
  238. [KMZT87a]
    C. Knessel, B.J. Matkowsky, Z. Schuss and C. Tier. Asymptotic expansions for a closed multiple access system. SIAM J. Comput., 16(2):378–398, 1987.MathSciNetGoogle Scholar
  239. [KMZT87b]
    C. Knessel, B.J. Matkowsky, Z. Schuss and C. Tier. A Markov-modulated M/G/1 queue I: stationary distribution. Queueing Syst., 1(4):355–374, 1987.Google Scholar
  240. [KMZT87c]
    C. Knessel, B.J. Matkowsky, Z. Schuss and C. Tier. The two repairmen problem: a finite source M/G/2 queue. SIAM J. Appl. Math., 47(2):367–397, 1987.MathSciNetGoogle Scholar
  241. [KORVE07]
    R.V. Kohn, F. Otto, M.G. Reznikoff, and E. Vanden-Eijnden. Action minimization and sharp-interface limits for the stochastic Allen–Cahn equation. Comm. Pure Appl. Math., 60(3):393–438, 2007.zbMATHMathSciNetGoogle Scholar
  242. [KOV89]
    C. Kipnis, S. Olla, and S.R.S. Varadhan. Hydrodynamics and large deviation for simple exclusion processes. Comm. Pure. Appl. Math., 42(2):115–137, 1989.zbMATHMathSciNetGoogle Scholar
  243. [KP92]
    Y.M. Kabanov and S.M. Pergamenshchikov. Singular perturbations of stochastic differential equations. Math. USSR-Sbor., 71:15–27, 1992.zbMATHMathSciNetGoogle Scholar
  244. [KP95]
    Y.M. Kabanov and S.M. Pergamenshchikov. Large deviations for solutions of singularly perturbed stochastic differential equations. Russ. Math. Surv., 50(5):989–1013, 1995.zbMATHMathSciNetGoogle Scholar
  245. [KP03a]
    Y. Kabanov and S. Pergamenshchikov. Two-Scale Stochastic Systems. Springer, 2003.Google Scholar
  246. [KPK13]
    S. Krumscheid, G.A. Pavliotis and S. Kalliadasis. Semiparametric drift and diffusion estimation for multiscale diffusions. Multiscale Model. Simul., 11(2):442–473, 2013.zbMATHMathSciNetGoogle Scholar
  247. [KPLM13]
    I.A. Khovanov, A.V. Polovinkin, D.G. Luchinsky, and P.V.E. McClintock. Noise-induced escape in an excitable system. Phys. Rev. E, 87:032116, 2013.Google Scholar
  248. [KPS91]
    Y.M. Kabanov, S.M. Pergamenshchikov, and J.M. Stoyanov. Asymptotic expansions for singularly perturbed stochastic differential equations. In V. Sazanov and T. Shervashidze, editors, New Trends in Probability and Statistics - In Honor of Yu. Prohorov, pages 413–435. VSP, 1991.Google Scholar
  249. [Kra40]
    H.A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7(4):284–304, 1940.zbMATHMathSciNetGoogle Scholar
  250. [KTY09]
    V. Krishnamurthy, K. Topley, and G. Yin. Consensus formation in a two-time-scale Markovian system. Multiscale Model. Simul., 7(4):1898–1927, 2009.zbMATHMathSciNetGoogle Scholar
  251. [Kue12a]
    C. Kuehn. Deterministic continuation of stochastic metastable equilibria via Lyapunov equations and ellipsoids. SIAM J. Sci. Comp., 34(3):A1635–A1658, 2012.zbMATHMathSciNetGoogle Scholar
  252. [Kue12b]
    C. Kuehn. Time-scale and noise optimality in self-organized critical adaptive networks. Phys. Rev. E, 85(2):026103, 2012.Google Scholar
  253. [Kus90]
    H.J. Kushner. Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Birkhäuser, 1990.Google Scholar
  254. [Kus99]
    R. Kuske. Probability densities for noisy delay bifurcation. J. Stat. Phys., 96(3):797–816, 1999.zbMATHMathSciNetGoogle Scholar
  255. [Kus00]
    R. Kuske. Gradient-particle solutions of Fokker–Planck equations for noisy delay bifurcations. SIAM J. Sci. Comput., 22(1):351–367, 2000.zbMATHMathSciNetGoogle Scholar
  256. [Kus03]
    R. Kuske. Multi-scale analysis of noise-sensitivity near a bifurcation. In IUTAM Symposium on Nonlinear Stochastic Dynamics, pages 147–156. Springer, 2003.Google Scholar
  257. [Kus10]
    R. Kuske. Competition of noise sources in systems with delay: the role of multiple time scales. J. Vibration and Control, 16(7):983–1003, 2010.zbMATHMathSciNetGoogle Scholar
  258. [Kwa13]
    F. Kwasniok. Analysis and modelling of glacial climate transitions using simple dynamical systems. Phil. Trans. R. Soc. A, 371(1991):20110472, 2013.Google Scholar
  259. [KY96a]
    R.Z. Khasminskii and G. Yin. Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains. SIAM J. Appl. Math., 56(1):277–293, 1996.zbMATHMathSciNetGoogle Scholar
  260. [KY96b]
    R.Z. Khasminskii and G. Yin. Asymptotic series for singularly perturbed Kolmogorov–Fokker–Planck equations. SIAM J. Appl. Math., 56(6):1766–1793, 1996.zbMATHMathSciNetGoogle Scholar
  261. [KY96c]
    R.Z. Khasminskii and G. Yin. On transition densities of singularly perturbed diffusions with fast and slow components. SIAM J. Appl. Math., 56(6):1794–1819, 1996.zbMATHMathSciNetGoogle Scholar
  262. [KY97]
    R.Z. Khasminskii and G. Yin. Constructing asymptotic series for probability distributions of Markov chains with weak and strong interactions. Quart. Appl. Math., 55(1):177–200, 1997.zbMATHMathSciNetGoogle Scholar
  263. [KY99]
    R.Z. Khasminskii and G. Yin. Singularly perturbed switching diffusions: rapid switchings and fast diffusions. J. Optim. Theor. Appl., 102(3):555–591, 1999.zbMATHMathSciNetGoogle Scholar
  264. [KY04]
    R.Z. Khasminskii and G. Yin. On averaging principles: an asymptotic expansion approach. SIAM J. Math. Anal., 35(6):1534–1560, 2004.zbMATHMathSciNetGoogle Scholar
  265. [KY05]
    R.Z. Khasminskii and G. Yin. Limit behavior of two-time-scale diffusions revisited. J. Differential Equat., 212(1):85–113, 2005.zbMATHMathSciNetGoogle Scholar
  266. [Lan08]
    P. Langevin. Sur la théorie du mouvement brownien. Compte-rendus des séances de l’Académie des sciences, 146:530–534, 1908.zbMATHGoogle Scholar
  267. [LC98]
    A. Longtin and D.R. Chialvo. Stochastic and deterministic resonances for excitable systems. Phys. Rev. Lett., 81(18):4012–4015, 1998.Google Scholar
  268. [LGONSG04]
    B. Lindner, J. Garcia-Ojalvo, A. Neiman, and L. Schimansky-Geier. Effects of noise in excitable systems. Physics Reports, 392:321–424, 2004.Google Scholar
  269. [Li08]
    X.M. Li. An averaging principle for a completely integrable stochastic Hamiltonian system. Nonlinearity, 21(4):803–822, 2008.zbMATHMathSciNetGoogle Scholar
  270. [Lin04]
    B. Lindner. Interspike interval statistics of neurons driven by colored noise. Phys. Rev. E, 69:022901, 2004.Google Scholar
  271. [Lip96]
    R. Liptser. Large deviations for two scaled diffusions. Prob. Theor. Rel. Fields, 106:71–104, 1996.zbMATHMathSciNetGoogle Scholar
  272. [LL09a]
    C. Laing and G. Lord, editors. Stochastic Methods in Neuroscience. OUP, 2009.Google Scholar
  273. [LL13]
    S. Lahbabi and F. Legoll. Effective dynamics for a kinetic Monte-Carlo model with slow and fast time scales. arXiv:1301.0266v1, pages 1–44, 2013.Google Scholar
  274. [LLB13]
    J. Li, K. Lu, and P. Bates. Normally hyperbolic invariant manifolds for random dynamical systems: Part I - persistence. Trans. Amer. Math. Soc., 365(11):5933–5966, 2013.zbMATHMathSciNetGoogle Scholar
  275. [LMYZ12]
    J. Lei, M.C. Mackey, R. Yvinec, and C. Zhuge. Adiabatic reduction of a piecewise deterministic Markov model of stochastic gene expression with bursting transcription. arXiv:1202.5411, pages 1–39, 2012.Google Scholar
  276. [Lon97]
    A. Longtin. Autonomous stochastic resonance in bursting neurons. Phys. Rev. E, 55(1):868–876, 1997.Google Scholar
  277. [LP93]
    G.D. Lythe and M.R.E. Proctor. Noise and slow–fast dynamics in a three-wave resonance problem. Phys. Rev. E, 47:3122–3127, 1993.Google Scholar
  278. [LS00]
    R. Liptser and V. Spokoiny. On estimating a dynamic function of a stochastic system with averaging. Stat. Inf. Stoch. Proc., 3(3):225–249, 2000.zbMATHMathSciNetGoogle Scholar
  279. [LSG99]
    B. Lindner and L. Schimansky-Geier. Analytical approach to the stochastic FitzHugh–Nagumo system and coherence resonance. Phys. Rev. E, 60(6):7270–7276, 1999.Google Scholar
  280. [LSG00]
    B. Lindner and L. Schimansky-Geier. Coherence and stochastic resonance in a two-state system. Phys. Rev. E, 61(6):6103–6110, 2000.Google Scholar
  281. [LSR07]
    T. Lelièvre, G. Stoltz, and M. Rousset. Computation of free energy profiles with parallel adaptive dynamics. J. Chem. Phys., 126:134111, 2007.Google Scholar
  282. [LSR10]
    T. Lelièvre, G. Stoltz, and M. Rousset. Free Energy Computations: A Mathematical Perspective. World Scientific, 2010.Google Scholar
  283. [Lyt95]
    G.D. Lythe. Noise and dynamic transitions. In Stochastic Partial Differential Equations (Edinburgh, 1994), pages 181–188. Springer, 1995.Google Scholar
  284. [Lyt96]
    G.D. Lythe. Domain formation in transitions with noise and a time-dependent bifurcation parameter. Phys. Rev. E, 53:4271, 1996.Google Scholar
  285. [MAG05]
    A.J. Majda, R.V. Abramov, and M.J. Grote. Information Theory and Stochastics for Multiscale Nonlinear Systems. AMS, 2005.Google Scholar
  286. [Mat95]
    P. Mathieu. Spectra, exit times and long time asymptotics in the zero-white-noise limit. Stoch. Stoch. Rep., 55(1):1–20, 1995.zbMATHMathSciNetGoogle Scholar
  287. [MBK13]
    W.W. Mohammed, D. Blömker, and K. Klepel. Modulation equation for stochastic Swift–Hohenberg equation. SIAM J. Math. Anal., 45(1):14–30, 2013.zbMATHMathSciNetGoogle Scholar
  288. [MGB95]
    F. Marchesoni, L. Gammaitoni, and A.R. Bulsara. Spatiotemporal stochastic resonance in a ϕ 4 model of kink-antikink nucleation. Phys. Rev. Lett., 76(15):2609, 1995.Google Scholar
  289. [Mil06]
    P.D. Miller. Applied Asymptotic Analysis. AMS, 2006.Google Scholar
  290. [MK93a]
    G.B. Di Masi and Y.M. Kabanov. The strong convergence of two-scale stochastic systems and singular perturbations of filtering equations. J. Math. Syst. Est. Contr., 3:207–224, 1993.zbMATHGoogle Scholar
  291. [MOS89]
    F. Martinelli, E. Olivieri, and E. Scoppola. Small random perturbations of finite-and infinite-dimensional dynamical systems: unpredictability of exit times. J. Stat. Phys., 55(3):477–504, 1989.zbMATHMathSciNetGoogle Scholar
  292. [MP11]
    A. Mokkadem and M. Pelletier. A generalization of the averaging procedure: the use of two-time-scale algorithms. SIAM J. Control Optim., 49(4):1523–1543, 2011.zbMATHMathSciNetGoogle Scholar
  293. [MS77]
    B.J. Matkowsky and Z. Schuss. The exit problem for randomly perturbed dynamical systems. SIAM J. Appl. Math., 33(2):365–382, 1977.zbMATHMathSciNetGoogle Scholar
  294. [MS79]
    B.J. Matkowsky and Z. Schuss. The exit problem: a new approach to diffusion across potential barriers. SIAM J. Appl. Math., 36(3):604–623, 1979.zbMATHMathSciNetGoogle Scholar
  295. [MS82]
    B.J. Matkowsky and Z. Schuss. Diffusion across characteristic boundaries. SIAM J. Appl. Math., 42(4):822–834, 1982.zbMATHMathSciNetGoogle Scholar
  296. [MS88]
    B.J. Matkowsky and Z. Schuss. Uniform asymptotic expansions in dynamical systems driven by colored noise. Phys. Rev. A, 38(5):2605, 1988.Google Scholar
  297. [MS92]
    R.S. Maier and D.L. Stein. Transition-rate theory for nongradient drift fields. Phys. Rev. Lett., 69(26):3691, 1992.Google Scholar
  298. [MS93a]
    R.S. Maier and D.L. Stein. Effect of focusing and caustics on exit phenomena in systems lacking detailed balance. Phys. Rev. Lett., 71(12):1783, 1993.Google Scholar
  299. [MS93b]
    R.S. Maier and D.L. Stein. Escape problem for irreversible systems. Phys. Rev. E, 48(2):931, 1993.Google Scholar
  300. [MS96a]
    R.S. Maier and D.L. Stein. Oscillatory behavior of the rate of escape through an unstable limit cycle. Phys. Rev. Lett., 77(24):4860, 1996.Google Scholar
  301. [MS96b]
    R.S. Maier and D.L. Stein. A scaling theory of bifurcations in the symmetric weak-noise escape problem. J. Stat. Phys., 83(3):291–357, 1996.zbMATHMathSciNetGoogle Scholar
  302. [MS97]
    R.S. Maier and D.L. Stein. Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math., 57(3):752–790, 1997.zbMATHMathSciNetGoogle Scholar
  303. [MS01a]
    R.S. Maier and D.L. Stein. Noise-activated escape from a sloshing potential well. Phys. Rev. Lett., 86(18):3942, 2001.Google Scholar
  304. [MSBJ82]
    B.J. Matkowsky, Z. Schuss, and E. Ben-Jacob. A singular perturbation approach to Kramers’ diffusion problem. SIAM J. Appl. Math., 42(4):835–849, 1982.zbMATHMathSciNetGoogle Scholar
  305. [MSK+84]
    B.J. Matkowsky, Z. Schuss, C. Knessl, C. Tier, and M. Mangel. Asymptotic solution of the Kramers–Moyal equation and first-passage times for Markov jump processes. Phys. Rev. A, 29(6):3359, 1984.Google Scholar
  306. [MST83]
    B.J. Matkowsky, Z. Schuss, and C. Tier. Diffusion across characteristic boundaries with critical points. SIAM J. Appl. Math., 43(4):673–695, 1983.zbMATHMathSciNetGoogle Scholar
  307. [MSVE06]
    P. Metzner, C. Schütte, and E. Vanden-Eijnden. Illustration of transition path theory on a collection of simple examples. J. Chem. Phys., 125:084110, 2006.Google Scholar
  308. [MSVE09]
    P. Metzner, C. Schütte, and E. Vanden-Eijnden. Transition path theory for Markov jump processes. Multiscale Model. Simul., 7(3):1192–1219, 2009.zbMATHGoogle Scholar
  309. [MT12]
    S. Méléard and V.C. Tran. Slow and fast scales for superprocess limits of age-structured populations. Stochastic Process. Appl., 122:250–276, 2012.zbMATHMathSciNetGoogle Scholar
  310. [MTVE99]
    A.J. Majda, I. Timofeyev, and E. Vanden-Eijnden. Models for stochastic climate prediction. Proc. Nat. Acad. USA, 96:14687–14691, 1999.zbMATHMathSciNetGoogle Scholar
  311. [MTVE01]
    A.J. Majda, I. Timofeyev, and E. Vanden-Eijnden. A mathematical framework for stochastic climate models. Comm. Pure and Appl. Math., 54:891–974, 2001.zbMATHMathSciNetGoogle Scholar
  312. [MTVE02]
    A.J. Majda, I. Timofeyev, and E. Vanden-Eijnden. A priori tests of a stochastic mode reduction strategy. Physica D, 170:206–252, 2002.zbMATHMathSciNetGoogle Scholar
  313. [MTVE03]
    A.J. Majda, I. Timofeyev, and E. Vanden-Eijnden. Systematic strategies for stochastic mode reduction in climate. J. Atmosph. Sci., 60:1705–1722, 2003.MathSciNetGoogle Scholar
  314. [MTVE06]
    A.J. Majda, I. Timofeyev, and E. Vanden-Eijnden. Stochastic models for selected slow variables in large deterministic systems. Nonlinearity, 19:769–794, 2006.zbMATHMathSciNetGoogle Scholar
  315. [Mun79]
    T. Munakata. Hydrodynamic equations from Fokker–Planck equations - multiple time scale method. J. Phys. Soc. Japan, 46:748–755, 1979.MathSciNetGoogle Scholar
  316. [MVE08]
    C.B. Muratov and E. Vanden-Eijnden. Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle. Chaos, 18:015111, 2008.MathSciNetGoogle Scholar
  317. [MVEE05]
    C.B. Muratov, E. Vanden-Eijnden, and W. E. Self-induced stochastic resonance in excitable systems. Physica D, 210:227–240, 2005.Google Scholar
  318. [Nam90]
    S. Namachchivaya. Stochastic bifurcation. Appl. Math. Comp., 38:101–159, 1990.zbMATHGoogle Scholar
  319. [Nar87]
    K. Narita. Asymptotic analysis for interactive oscillators of the van der Pol type. Adv. Appl. Prob., 19:44–80, 1987.zbMATHMathSciNetGoogle Scholar
  320. [Nar91]
    K. Narita. Asymptotic behavior of velocity process in the Smoluchowski-Kramers approximation for stochastic differential equations. Adv. Appl. Prob., 23:317–326, 1991.zbMATHMathSciNetGoogle Scholar
  321. [Nar93]
    K. Narita. Asymptotic behavior of solutions of SDE for relaxation oscillations. SIAM J. Math. Anal., 24(1):172–199, 1993.zbMATHMathSciNetGoogle Scholar
  322. [NBK13]
    J.M. Newby, P.C. Bressloff, and J.P. Keener. Breakdown of fast–slow analysis in an excitable system with channel noise. Phys. Rev. Lett., 111(12):128101, 2013.Google Scholar
  323. [NKMS90]
    T. Naeh, M.M. Klosek, B.J. Matkowsky, and Z. Schuss. A direct approach to the exit problem. SIAM J. Appl. Math., 50(2):595–627, 1990.zbMATHMathSciNetGoogle Scholar
  324. [NN81]
    C. Nicolis and G. Nicolis. Stochastic aspects of climatic transitions—additive fluctuations. Tellus, 33(3):225–234, 1981.MathSciNetGoogle Scholar
  325. [NN13]
    F. Noé and F. Nüske. A variational approach to modeling slow processes in stochastic dynamical systems. Multiscale Model. Simul., 11(2):635–655, 2013.zbMATHMathSciNetGoogle Scholar
  326. [Nor06]
    J.R. Norris. Markov Chains. Cambridge University Press, 2006.Google Scholar
  327. [Num84]
    E. Nummelin. General irreducible Markov chains and nonnegative operators, volume 84 of Tracts in Mathematics. CUP, 1984.Google Scholar
  328. [NY10]
    S.L. Nguyen and G. Yin. Asymptotic properties of Markov-modulated random sequences with fast and slow timescales. Stochastics, 82(4):445–474, 2010.zbMATHMathSciNetGoogle Scholar
  329. [O’B03]
    N. O’Bryant. A noisy system with a flattened Hamiltonian and multiple time scales. Stoch. Dyn., 3(1):1–54, 2003.zbMATHMathSciNetGoogle Scholar
  330. [Øks03]
    B. Øksendal. Stochastic Differential Equations. Springer, Berlin Heidelberg, Germany, 5th edition, 2003.Google Scholar
  331. [OP11]
    M. Ottobre and G.A. Pavliotis. Asymptotic analysis for the generalized Langevin equation. Nonlinearity, 24(5):1629, 2011.Google Scholar
  332. [OS95]
    E. Olivieri and E. Scoppola. Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case. J. Stat. Phys., 79(3):613–647, 1995.Google Scholar
  333. [OS96]
    E. Olivieri and E. Scoppola. Markov chains with exponentially small transition probabilities: first exit problem from a general domain. II. The general case. J. Stat. Phys., 84(5):987–1041, 1996.Google Scholar
  334. [OV05]
    E. Olivieri and M.E. Vares. Large Deviations and Metastability. CUP, 2005.Google Scholar
  335. [Pap76]
    G.C. Papanicolaou. Some probabilistic problems and methods in singular perturbations. Rocky Mountain J. Math., 6:653–674, 1976.zbMATHMathSciNetGoogle Scholar
  336. [Pap77]
    G.C. Papanicolaou. Introduction to the asymptotic analysis of stochastic equations. In Modern Modeling of Continuum Phenomena, volume 16 of Lect. Appl. Math., pages 109–147. AMS, 1977.Google Scholar
  337. [Pav05]
    G.A. Pavliotis. A multiscale approach to Brownian motors. Phys. Lett. A, 344(5):331–345, 2005.zbMATHGoogle Scholar
  338. [Per94a]
    S. Pergamenshchikov. Asymptotic expansions for a model with distinguished ‘fast’ and ‘slow’ variables, described by a system of singularly perturbed stochastic differential equations. Russ. Math. Surv., 49(4):1–44, 1994.zbMATHMathSciNetGoogle Scholar
  339. [PK75]
    G.C. Papanicolaou and W. Kohler. Asymptotic analysis of deterministic and stochastic equations with rapidly varying components. Comm. Math. Phys., 45(3):217–232, 1975.zbMATHMathSciNetGoogle Scholar
  340. [PK97]
    A.S. Pikovsky and J. Kurths. Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett., 78:775–778, 1997.zbMATHMathSciNetGoogle Scholar
  341. [PP13]
    P. Pfaffelhuber and L. Popovic. Scaling limits of spatial chemical reaction networks. arXiv:1302.0774v1, pages 1–50, 2013.Google Scholar
  342. [PPS09]
    A. Papavasiliou, G.A. Pavliotis, and A.M. Stuart. Maximum likelihood drift estimation for multiscale diffusions. Stoch. Proc. Appl., 119(10):3173–3210, 2009.zbMATHMathSciNetGoogle Scholar
  343. [Pro05]
    P. Protter. Stochastic Integration and Differential Equations - Version 2.1. Springer, 2005.Google Scholar
  344. [PS05a]
    G.A. Pavliotis and A.M. Stuart. Analysis of white noise limits for stochastic systems with two fast relaxation times. Multiscale Model. Simul., 4(1):1–35, 2005.zbMATHMathSciNetGoogle Scholar
  345. [PS05b]
    G.A. Pavliotis and A.M. Stuart. Periodic homogenization for inertial particles. Physica D, 204(3): 161–187, 2005.zbMATHMathSciNetGoogle Scholar
  346. [PS06]
    V.A. Pliss and G.R. Sell. Averaging methods for stochastic dynamics of complex reaction networks: description of multiscale couplings. Multiscale Model. Simul., 5(2):497–513, 2006.MathSciNetGoogle Scholar
  347. [PS07a]
    G.A. Pavliotis and A.M. Stuart. Parameter estimation for multiscale diffusions. J. Stat. Phys., 127(4):741–781, 2007.zbMATHMathSciNetGoogle Scholar
  348. [PSV10]
    M.A. Peletier, G. Savaré, and M. Veneroni. From diffusion to reaction via Γ-convergence. SIAM J. Math. Anal., 42(4):1805–1825, 2010.zbMATHMathSciNetGoogle Scholar
  349. [PSV12]
    M.A. Peletier, G. Savaré, and M. Veneroni. Chemical reactions as Γ-limit of diffusion. SIAM Rev., 54(2):327–352, 2012.zbMATHMathSciNetGoogle Scholar
  350. [PSZ07]
    G.A. Pavliotis, A.M. Stuart, and K.C. Zygalakis. Homogenization for inertial particles in a random flow. Comm. Math. Sci., 5(3):506–531, 2007.MathSciNetGoogle Scholar
  351. [PTK+11]
    M. Pradas, D. Tseluiko, S. Kalliadasis, D.T. Papageorgiou, and G.A. Pavliotis. Noise induced state transitions, intermittency, and universality in the noisy Kuramoto-Sivashinksy equation. Phys. Rev. Lett., 106(6):060602, 2011.Google Scholar
  352. [Puh13]
    A.A. Puhalskii. Large deviations of coupled diffusions with time scale separation. arXiv:1306.5446v1, pages 1–74, 2013.Google Scholar
  353. [PV01]
    E. Pardoux and Yu. Veretennikov. On the Poisson equation and diffusion approximation. I. Ann. Probab., 29(3):1061–1085, 2001.Google Scholar
  354. [PWPK10]
    S. Pillay, M.J. Ward, A. Peirce, and T. Kolokolnikov. An asymptotic analysis of the mean first passage time for narrow escape problems: Part I Two-dimensional domains. Multiscale Mod. Simul., 8(3):803–835, 2010.zbMATHMathSciNetGoogle Scholar
  355. [PZ92]
    G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, 1992.Google Scholar
  356. [Raz78]
    V.D. Razevig. Reduction of stochastic differential equations with small parameters and stochastic integrals. Int. J. Contr., 28(5):707–720, 1978.zbMATHMathSciNetGoogle Scholar
  357. [RD13]
    J. Ren and J. Duan. A parameter estimation method based on random slow manifolds. arXiv:1303.4600, pages 1–16, 2013.Google Scholar
  358. [RDJ12]
    J. Ren, J. Duan, and C.K.R.T. Jones. Approximation of random slow manifolds and settling of inertial particles under uncertainty. arXiv:1212.4216v1, pages 1–26, 2012.Google Scholar
  359. [Res92]
    S.I. Resnick. Adventures in Stochastic Processes. Birkhäuser, 1992.Google Scholar
  360. [Ris96]
    H. Risken. The Fokker–Planck Equation. Springer, 1996.Google Scholar
  361. [RMSaSL12]
    V.M. Rozenbaum, Yu.A. Makhnovskii, I.V. Shapochkina S.-Y. Sheu, D.-Y. Yamg and S.H. Lin. Adiabatically slow and adiabatically fast driven ratchets. Phys. Rev. E, 85:041116, 2012.Google Scholar
  362. [Ros06]
    S. Ross. A First Course in Probability. Pearson Prentice Hall, 2006.Google Scholar
  363. [RW88]
    J.R. Rohlicek and A.S. Willsky. Multiple time scale decomposition of discrete time Markov chains. Syst. Contr. Lett., 11(4):309–314, 1988.zbMATHMathSciNetGoogle Scholar
  364. [Sch80]
    Z. Schuss. Singular perturbation methods in stochastic differential equations of mathematical physics. SIAM Rev., 22(2):119–155, 1980.zbMATHMathSciNetGoogle Scholar
  365. [Sch09b]
    Z. Schuss. Theory and Applications of Stochastic Processes: An Analytical Approach. Springer, 2009.Google Scholar
  366. [SGH01]
    G. Schmid, I. Goychuk, and P. Hänggi. Stochastic resonance as a collective property of ion channel assemblies. Europhys. Lett., 56(1):22, 2001.Google Scholar
  367. [SH96]
    K.R. Schenk-Hoppé. Bifurcation scenarios of the noisy Duffing–van der Pol oscillator. Nonlinear Dyn., 11:255–274, 1996.Google Scholar
  368. [SH98]
    K.R. Schenk-Hoppé. Random attractors - general properties, existence and applications to stochastic bifurcation theory. Discr. Cont. Dyn. Syst. A, 4(1):99–130, 1998.zbMATHGoogle Scholar
  369. [Sim83]
    B. Simon. Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymptotic expansions. Ann. IHP (A) Phys. Théor., 38(3):295–308, 1983.Google Scholar
  370. [Sim84]
    B. Simon. Semiclassical analysis of low lying eigenvalues, II. Tunneling. Ann. Math., 120:89–118, 1984.Google Scholar
  371. [SK11b]
    D.J.W. Simpson and R. Kuske. Mixed-mode oscillations in a stochastic, piecewise-linear system. Physica D, 240(14):1189–1198, 2011.zbMATHGoogle Scholar
  372. [SM11]
    T. Schäfer and R.O. Moore. A path integral method for coarse-graining noise in stochastic differential equations with multiple time scales. Physica D, 240:89–97, 2011.zbMATHMathSciNetGoogle Scholar
  373. [SMSG07]
    M. Sieber, H. Malchow, and L. Schimansky-Geier. Constructive effects of environmental noise in an excitable prey–predator plankton system with infected prey. Ecol. Complex., 4(4):223–233, 2007.Google Scholar
  374. [SN13]
    D.L. Stein and C.M. Newman. Rugged landscapes and timescale distributions in complex systems. In Emergence, Complexity, and Computation. Springer, 2013. to appear.Google Scholar
  375. [Sow01]
    R.B. Sowers. On the tangent flow of a stochastic differential equation with fast drift. Trans. Amer. Math. Soc., 353(4):1321–1334, 2001.zbMATHMathSciNetGoogle Scholar
  376. [Sow02]
    R.B. Sowers. Stochastic averaging with a flattened Hamiltonian: a Markov process on a stratified space (a whiskered sphere). Trans. Amer. Math. Soc., 354(3):853–900, 2002.zbMATHMathSciNetGoogle Scholar
  377. [Sow08]
    R.B. Sowers. Random perturbations of canards. J. Theor. Probab., 21:824–889, 2008.zbMATHMathSciNetGoogle Scholar
  378. [Spi13a]
    K. Spiliopoulos. Fluctuation analysis and short time asymptotics for multiple scales diffusion processes. arXiv:1306:1499v1, pages 1–18, 2013.Google Scholar
  379. [Spi13b]
    K. Spiliopoulos. Large deviations and importance sampling for systems of slow–fast motion. Appl. Math. Optim., 67(1):123–161, 2013.zbMATHMathSciNetGoogle Scholar
  380. [SRT04]
    J. Su, J. Rubin, and D. Terman. Effects of noise on elliptic bursters. Nonlinearity, 17:133–157, 2004.zbMATHMathSciNetGoogle Scholar
  381. [SS08b]
    B. Schmalfuss and K.R. Schneider. Invariant manifolds for random dynamical systems with slow and fast variables. J. Dyn. Diff. Eq., 20(1):133–164, 2008.zbMATHMathSciNetGoogle Scholar
  382. [SS11a]
    A.M. Samoilenko and O. Stanzhytskyi. Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations. World Scientific, 2011.Google Scholar
  383. [SS11b]
    M. Sarich and C. Schütte. Approximating selected non-dominant timescales by Markov state models. Comm. Math. Sci., 10(3):1001–1013, 2011.Google Scholar
  384. [SSH06a]
    A. Singer, Z. Schuss, and D. Holcman. Narrow escape, part II: The circular disk. J. Stat. Phys., 122(3):465–489, 2006.zbMATHMathSciNetGoogle Scholar
  385. [SSH06b]
    A. Singer, Z. Schuss, and D. Holcman. Narrow escape, part III: Non-smooth domains and Riemann surfaces. J. Stat. Phys., 122(3):491–509, 2006.zbMATHMathSciNetGoogle Scholar
  386. [SSH07]
    Z. Schuss, A. Singer, and D. Holcman. The narrow escape problem for diffusion in cellular microdomains. Proc. Natl. Acad. Sci. USA, 104(41):16098–16103, 2007.Google Scholar
  387. [SSH08]
    A. Singer, Z. Schuss, and D. Holcman. Narrow escape and leakage of brownian particles. Phys. Rev. E, 78(5):051111, 2008.Google Scholar
  388. [SSHE06]
    A. Singer, Z. Schuss, D. Holcman, and R.S. Eisenberg. Narrow escape, part I. J. Stat. Phys., 122(3):437–463, 2006.zbMATHMathSciNetGoogle Scholar
  389. [Ste04]
    D.L. Stein. Critical behavior of the Kramers escape rate in asymmetric classical field theories. J. Stat. Phys., 114(5):1537–1556, 2004.zbMATHGoogle Scholar
  390. [Sug95]
    M. Sugiura. Metastable behaviors of diffusion processes with small parameter. J. Math. Soc. Japan, 47(4):755–788, 1995.zbMATHMathSciNetGoogle Scholar
  391. [SWHH04]
    C. Schütte, J. Walter, C. Hartmann, and W. Huisinga. An averaging principle for fast degrees of freedom exhibiting long-term correlations. Multiscale Model. Simul., 2(3):501–526, 2004.zbMATHMathSciNetGoogle Scholar
  392. [Tal99]
    P. Talkner. Stochastic resonance in the semiadiabatic limit. New J. Phys., 1(1):4, 1999.Google Scholar
  393. [TGS12]
    P. Thomas, R. Grima, and A.V. Straube. Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators. Phys. Rev. E, 86:041110, 2012.Google Scholar
  394. [TGT95]
    D.N.C. Tse, R.G. Gallager, and J.N. Tsitsiklis. Statistical multiplexing of multiple time-scale Markov streams. IEEE J. Selected Areas in Comm., 13(6):1028–1038, 1995.Google Scholar
  395. [TKD13]
    J. Touboul, M. Krupa, and M. Desroches. Noise-induced canard and mixed-mode oscillations in large stochastic networks with multiple timescales. arXiv:1302:7159v1, pages 1–22, 2013.Google Scholar
  396. [TM88]
    M.C. Torrent and M. San Miguel. Stochastic-dynamics characterization of delayed laser threshold instability with swept control parameter. Phys. Rev. A, 38(1):245–251, 1988.Google Scholar
  397. [TR98]
    H.C. Tuckwell and R. Rodriguez. Analytical and simulation results for stochastic Fitzhugh-Nagumo neurons and neural networks. J. Comput. Neurosci., 5(1):91–113, 1998.zbMATHGoogle Scholar
  398. [TSG11]
    P. Thomas, A.V. Straube, and R. Grima. Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks. J. Chem. Phys., 135(18):181103, 2011.Google Scholar
  399. [Var84]
    S.R.S. Varadhan. Large Deviations and Applications. SIAM, 1984.Google Scholar
  400. [Var08]
    S.R.S. Varadhan. Large deviations. Ann. Probab., 36(2):397–419, 2008.zbMATHMathSciNetGoogle Scholar
  401. [vK07]
    N.G. van Kampen. Stochastic Processes in Physics and Chemistry. North-Holland, 2007.Google Scholar
  402. [VMS00]
    S. Varela, C. Masoller, and A.C. Sicardi. Numerical simulations of the effect of noise on a delayed pitchfork bifurcation. Physica A, 283(1):228–232, 2000.Google Scholar
  403. [VS00]
    G. De Vries and A. Sherman. Channel sharing in pancreatic-β-cells revisited: enhancement of emergent bursting by noise. J. Theor. Biol., 207(4):513–530, 2000.Google Scholar
  404. [Wai11]
    G. Wainrib. Noise-controlled dynamics through the averaging principle for stochastic slow–fast systems. Phys. Rev. E, 84:051113, 2011.Google Scholar
  405. [WB87a]
    D. Wycoff and N.L. Balazs. Multiple time scales analysis for the Kramers-Chandrasekhar equation. Phys. A, 146:175–200, 1987.MathSciNetGoogle Scholar
  406. [WB87b]
    D. Wycoff and N.L. Balazs. Multiple time scales analysis for the Kramers-Chandrasekhar equation with a weak magnetic field. Phys. A, 146:201–218, 1987.MathSciNetGoogle Scholar
  407. [WB87c]
    D. Wycoff and N.L. Balazs. Separation of fast and slow variables for a linear system by the method of multiple time scales. Phys. A, 146:219–241, 1987.MathSciNetGoogle Scholar
  408. [WD07]
    W. Wang and J. Duan. Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions. Comm. Math. Phys., 275:163–186, 2007.zbMATHMathSciNetGoogle Scholar
  409. [WD09]
    W. Wang and J. Duan. Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions. Stoch. Anal. Appl., 27(3):431–459, 2009.zbMATHMathSciNetGoogle Scholar
  410. [WDD07]
    W. Wang, D. Cao, and J. Duan. Effective macroscopic dynamics of stochastic partial differential equations in perforated domains. SIAM J. Math. Anal., 38:1508–1527, 2007.zbMATHMathSciNetGoogle Scholar
  411. [WF70]
    A.D. Wentzell and M.I. Freidlin. On small random perturbations of dynamical systems. Russ. Marth. Surv., 25:1–55, 1970.Google Scholar
  412. [WF73]
    A.D. Wentzell and M.I. Freidlin. Some problems concerning stability under small random perturbations. Theory Probab. Appl., 17(2):269–283, 1973.Google Scholar
  413. [WHK93]
    M.J. Ward, W.D. Heshaw, and J.B. Keller. Summing logarithmic expansions for singularly perturbed eigenvalue problems. SIAM J. Appl. Math., 53(3):799–828, 1993.zbMATHMathSciNetGoogle Scholar
  414. [Wil91]
    D. Williams. Probability with Martingales. CUP, 1991.Google Scholar
  415. [WK93]
    M.J. Ward and J.B. Keller. Strong localized perturbations of eigenvalue problems. SIAM J. Appl. Math., 53(3):770–798, 1993.zbMATHMathSciNetGoogle Scholar
  416. [WR12]
    W. Wang and A.J. Roberts. Average and deviation for slow–fast stochastic partial differential equations. J. Differential Equat., 253(5):1265–1286, 2012.zbMATHMathSciNetGoogle Scholar
  417. [WR13]
    W. Wang and A.J. Roberts. Slow manifold and averaging for slow–fast stochastic differential system. J. Math. Anal. Appl., 398(2):822–839, 2013.zbMATHMathSciNetGoogle Scholar
  418. [WRD12]
    W. Wang, A.J. Roberts, and J. Duan. Large deviations and approximations for slow–fast stochastic reaction–diffusion equations. J. Differential Equat., 12:3501–3522, 2012.MathSciNetGoogle Scholar
  419. [WS06a]
    J. Walter and C. Schütte. Conditional averaging for diffusive fast–slow systems: a sketch for derivation. In Analysis, Modeling and Simulation of Multiscale Problems, pages 647–682. Springer, 2006.Google Scholar
  420. [WSB04]
    T. Wellens, V. Shatokhin, and A. Buchleitner. Stochastic resonance. Reports on Progress in Physics, 67:45–105, 2004.Google Scholar
  421. [WSGR11]
    J. Wang, J. Su, H. Perez Gonzalez, and J. Rubin. A reliability study of square wave bursting beta-cells with noise. Discr. Cont. Dyn. Syst. B, 16:569–588, 2011.zbMATHMathSciNetGoogle Scholar
  422. [WZY06]
    J.W. Wang, Q. Zhang, and G. Yin. Two-time-scale hybrid filters: near optimality. SIAM J. Contr. Optim., 45:298–319, 2006.MathSciNetGoogle Scholar
  423. [XDX11]
    Y. Xu, J. Duan, and W. Xu. An averaging principle for stochastic dynamical systems with Lévy noise. Physica D, 240(17):1395–1401, 2011.zbMATHMathSciNetGoogle Scholar
  424. [YD03]
    G. Yin and S. Dey. Weak convergence of hybrid filtering problems involving nearly completely decomposable hidden Markov chains. SIAM J. Contr. Optim., 41(6):1820–1842, 2003.zbMATHMathSciNetGoogle Scholar
  425. [Yin01]
    G. Yin. On limit results for a class of singularly perturbed switching diffusions. J. Theor. Prob., 14(3):673–697, 2001.zbMATHGoogle Scholar
  426. [YK99]
    G. Yin and M. Kniazeva. Singularly perturbed multidimensional switching diffusions with fast and slow switchings. J. Math. Anal. Appl., 229(2):605–630, 1999.zbMATHMathSciNetGoogle Scholar
  427. [YK05]
    G. Yin and V. Krishnamurthy. Least mean square algorithms with Markov regime-switching limit. IEEE Trans. Aut. Contr., 50(5):577–593, 2005.MathSciNetGoogle Scholar
  428. [YKL06]
    N. Yu, R. Kuske, and Y.X. Li. Stochastic phase dynamics: multiscale behavior and coherence measures. Phys. Rev. E, 73(5):056205, 2006.Google Scholar
  429. [YKL08]
    N. Yu, R. Kuske, and Y.X. Li. Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators. Chaos, 18:015112, 2008.MathSciNetGoogle Scholar
  430. [Yvi13]
    R. Yvinec. Adiabatic reduction of models of stochastic gene expression with bursting. arXiv:1301.1293v1, pages 1–24, 2013.Google Scholar
  431. [YY04]
    G. Yin and H. Yang. Two-time-scale jump-diffusion models with Markovian switching regimes. Stoch. Stoch. Rep., 76(2):77–99, 2004.zbMATHMathSciNetGoogle Scholar
  432. [YYYZ02]
    H. Yang, G. Yin, K. Yin, and Q. Zhang. Control of singularly perturbed Markov chains: a numerical study. ANZIAM J., 45:49–74, 2002.zbMATHMathSciNetGoogle Scholar
  433. [YZ94]
    G. Yin and Q. Zhang. Near optimality of stochastic control in systems with unknown parameter processes. Appl. Math. Optim., 29(3):263–284, 1994.zbMATHMathSciNetGoogle Scholar
  434. [YZ97]
    G. Yin and Q. Zhang. Control of dynamic systems under the influence of singularly perturbed Markov chains. J. Math. Anal. Appl., 216(1):343–367, 1997.zbMATHMathSciNetGoogle Scholar
  435. [YZ98]
    G.G. Yin and Q. Zhang. Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach. Springer, 1998.Google Scholar
  436. [YZ00]
    G. Yin and Q. Zhang. Singularly perturbed discrete-time Markov chains. SIAM J. Appl. Math., 61(3):834–854, 2000.zbMATHMathSciNetGoogle Scholar
  437. [YZ02a]
    G. Yin and H. Zhang. Countable-state-space Markov chains with two time scales and applications to queueing systems. Adv. Appl. Prob., 34(3):662–688, 2002.zbMATHMathSciNetGoogle Scholar
  438. [YZ02b]
    G. Yin and Q. Zhang. Hybrid singular systems of differential equations. Science China F, 45(4): 241–258, 2002.zbMATHMathSciNetGoogle Scholar
  439. [YZ03]
    G. Yin and Q. Zhang. Discrete-time singularly perturbed Markov chains. In Stochastic Modeling and Optimization, pages 1–42. Springer, 2003.Google Scholar
  440. [YZ04b]
    G. Yin and Q. Zhang. Two-time-scale Markov chains and applications to quasi-birth-death queues. SIAM J. Appl. Math., 65(2):567–586, 2004.MathSciNetGoogle Scholar
  441. [YZ05]
    G.G. Yin and Q. Zhang. Discrete-time Markov Chains: Two-Time-Scale Methods and Applications. Springer, 2005.Google Scholar
  442. [YZ07]
    G. Yin and Q. Zhang. Singularly perturbed Markov chains: limit results and applications. Ann. Appl. Prob., 17(1):207–229, 2007.zbMATHMathSciNetGoogle Scholar
  443. [YZ08]
    G. Yin and Q. Zhang. Discrete-time Markov chains with two-time scales and a countable state space: limit results and queueing applications. Stochastics, 80(4):339–369, 2008.zbMATHMathSciNetGoogle Scholar
  444. [YZ10]
    G.G. Yin and C. Zhu. Hybrid Switching Diffusions: Properties and Applications. Springer, 2010.Google Scholar
  445. [YZB00a]
    G. Yin, Q. Zhang, and G. Badowski. Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states. Ann. Appl. Math., 10(2):549–572, 2000.zbMATHMathSciNetGoogle Scholar
  446. [YZB00b]
    G. Yin, Q. Zhang, and G. Badowski. Occupation measures of singularly perturbed Markov chains with absorbing states. Acta Math. Sinica, 16(1):161–180, 2000.zbMATHMathSciNetGoogle Scholar
  447. [YZB00c]
    G. Yin, Q. Zhang, and G. Badowski. Singularly perturbed Markov chains: convergence and aggregation. J. Multivar. Anal., 72(2):208–229, 2000.zbMATHMathSciNetGoogle Scholar
  448. [ZGO13]
    Q. Zhang, J. Gong, and C.H. Oh. Intrinsic dynamical fluctuation assisted symmetry breaking in adiabatic following. Phys. Rev. Lett., 110:130402, 2013.Google Scholar
  449. [Zha95]
    Q. Zhang. Risk-sensitive production planning of stochastic manufacturing systems: a singular perturbation approach. SIAM J. Contr. Optim., 33(2):498–527, 1995.zbMATHGoogle Scholar
  450. [Zha96]
    Q. Zhang. Finite state Markovian decision processes with weak and strong interactions. Stochastics, 59(3):283–304, 1996.zbMATHMathSciNetGoogle Scholar
  451. [ZMdB89]
    H. Zeghlache, P. Mandel, and C. Van den Broeck. Influence of noise on delayed bifurcations. Phys. Rev. A, 40:286–294, 1989.Google Scholar
  452. [ZY96]
    Q. Zhang and G.G. Yin. A central limit theorem for singularly perturbed nonstationary finite state Markov chains. Ann. Appl. Prob., 6(2):650–670, 1996.zbMATHMathSciNetGoogle Scholar
  453. [ZY99]
    Q. Zhang and G.G. Yin. On nearly optimal controls of hybrid LQG problems. IEEE Trans. Aut. Contr., 44(12):2271–2282, 1999.zbMATHMathSciNetGoogle Scholar
  454. [ZY04]
    Q. Zhang and G.G. Yin. Exponential bounds for discrete-time singularly perturbed Markov chains. J. Math. Anal. Appl., 293(2):645–662, 2004.zbMATHMathSciNetGoogle Scholar
  455. [ZYB97]
    Q. Zhang, G.G. Yin, and E.K. Boukas. Controlled Markov chains with weak and strong interactions: asymptotic optimality and applications to manufacturing. J. Optim. Theor. Appl., 94(1):169–194, 1997.zbMATHMathSciNetGoogle Scholar
  456. [ZYL05]
    Q. Zhang, G.G. Yin, and R.H. Liu. A near-optimal selling rule for a two-time-scale market model. Multiscale Model. Simul., 4(1):172–193, 2005.zbMATHMathSciNetGoogle Scholar
  457. [ZYM07]
    Q. Zhang, G.G. Yin, and J.B. Moore. Two-time-scale approximation for Wonham filters. IEEE Trans. Inf. Theor., 53(5):1706–1715, 2007.MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Kuehn
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

Personalised recommendations