Technical Background

  • George William Albert ConstableEmail author
Part of the Springer Theses book series (Springer Theses)


In this chapter, I will review some of the fundamental ideas and mathematical tools employed in the analysis of stochastic systems.


Transition Rate Master Equation Stochastic Differential Equation Stochastic System Slow Manifold 
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  1. 1.
    L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics (Springer, Berlin, 2003)Google Scholar
  2. 2.
    L. Arnold, P. Imkeller, Normal forms for stochastic differential equations. Probab. Theory Relat. Fields 110, 559–588 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L. Arnold, X. Kedai, Simultaneous normal form and center manifold reduction for random differential equations, in Equadiff 91 : International Conference on Differential Equations, ed. by C. Perello, C. Sim, J. Sola-Morales (1991)Google Scholar
  4. 4.
    N. Berglund, B. Gentz, Geometric singular pertubation theory for stochastic differential equations. J. Differ. Equ. 191, 1–54 (2003)MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    R.A. Blythe, A.J. McKane, Stochastic models of evolution in genetics, ecology and linguistics. J. Stat. Mech. P07018 (2007)Google Scholar
  6. 6.
    X. Chao, A.J. Roberts, On the low dimensional modelling of Stratanovich stochastic differential equations. Phys. A 225, 62–80 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    M.N. Contou-Carrere, V. Sotiropoulos, Y.N. Kaznessis, P. Daoutidis, Model reduction of multi-scale chemical Langevin equations. Syst. Control Lett. 60, 75–86 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    P.H. Coullet, C. Elphick, E. Tirapegui, Normal form of Hopf bifurcation with noise. Phys. Lett. A 111, 277–282 (1985)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    J.F. Crow, M. Kimura, Some genetic problems in natural populations, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, ed. by J. Neyman, University of California Press, Berkeley (1956), pp. 1–22Google Scholar
  10. 10.
    W.J. Ewens, Population Genetics (Wilmer Brother Limited, Birkenhead, 1969)CrossRefzbMATHGoogle Scholar
  11. 11.
    W.J. Ewens, Mathematical Population Genetics, 2nd edn. (Springer, Berlin, 2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    R.A. Fisher, On the dominance ratio. Proc. Roy. Soc. Edinb. 42, 321–341 (1922)Google Scholar
  13. 13.
    R.A. Fisher, The Genetical Theory of Natural Selection (Clarendon Press, Oxford, 1930)CrossRefzbMATHGoogle Scholar
  14. 14.
    C.W. Gardiner, Adiabatic elimination in stochastic systems. I. Formulation of methods and application to few-variable systems. Phys. Rev. A 29, 2814–2822 (1984)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 2009)Google Scholar
  16. 16.
    D.T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    D.T. Gillespie, Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977)CrossRefGoogle Scholar
  18. 18.
    A.J.F. Griffiths, S.R. Wessler, R.C. Lewontin, S.B. Carroll, Introduction to Genetic Analysis, 9th edn. (W. H. Freeman and Company, New York, 2007)Google Scholar
  19. 19.
    H. Haken, Synergetics (Springer, Berlin, 1983)CrossRefGoogle Scholar
  20. 20.
    H. Haken, A. Wunderlin, Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps. Z. Phys. B 47, 179–187 (1982)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    D.L. Hartl, A.G. Clark, Principles of Population Genetics, 4th edn. Sinauer Associates Inc., Sunderland, Mass. (2007)Google Scholar
  22. 22.
    R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991)CrossRefzbMATHGoogle Scholar
  23. 23.
    R.Z. Khasminskii, A limit theorem for the solutions of differential equations with random right-hand sides. Theor. Probab. Appl. 11, 390–406 (1966)CrossRefGoogle Scholar
  24. 24.
    M. Kimura, Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harb. Symp. Quant. Biol. 20, 33–53 (1955)CrossRefGoogle Scholar
  25. 25.
    M. Kimura, Diffusion models in population genetics. J. Appl. Probab. 1, 177–232 (1964)CrossRefzbMATHGoogle Scholar
  26. 26.
    M. Kimura, Population Genetics, Molecular Evolution and the Neutral Theory (Chicago University Press, Chicago, 1994)Google Scholar
  27. 27.
    M. Kimura, G.H. Weiss, The stepping stone model of population structure and the decrease in genetic correlation with distance. Genetics 49, 561–576 (1964)Google Scholar
  28. 28.
    E. Knobloch, K.A. Wiesenfeld, Bifurcations in fluctuating systems: the center manifold approach. J. Stat. Phys. 33, 611–637 (1983)MathSciNetADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Y. Lan, T.C. Elston, G.A. Papoian, Elimination of fast variables in chemical Langevin equations. J. Chem. Phys. 129, 214115 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    M. Lax, Fluctuations from the nonequilibrium steady state. Rev. Mod. Phys. 32, 25–64 (1960)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Y.T. Lin, H. Kim, C.R. Doering, Features of fast living: on the weak selection for longevity in degenerate birth-death processes. J. Stat. Phys. 148, 646–662 (2012)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    T. Maruyama, Stochastic Problems in Population Genetics (Springer, Berlin, 1977)CrossRefzbMATHGoogle Scholar
  33. 33.
    A.J. McKane, T. Biancalani, T. Rogers, Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond. Bull. Math. Biol. 76, 895–921 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    R.E. Michod, Darwinian Dynamics: Evolutionary Transitions in Fitness and Individuality (Princeton University Press, New Jersey, 2000)Google Scholar
  35. 35.
    M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life (Harvard University Press, Cambridge, 2006)Google Scholar
  36. 36.
    G.A. Pavliotis, A.M. Stuart, Averaging and Homogenization, vol. 53, Multiscale Methods (Springer, New York, 2008)zbMATHGoogle Scholar
  37. 37.
    L.E. Reichl, A Modern Course in Statistical Physics (Wiley VCH, New York, 1998)zbMATHGoogle Scholar
  38. 38.
    H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1989)CrossRefzbMATHGoogle Scholar
  39. 39.
    A.J. Roberts, Normal form transforms seperate slow and fast modes in stochastic dynamics systems. Phys. A 387, 12–38 (2008)MathSciNetCrossRefGoogle Scholar
  40. 40.
    G. Schöner, H. Haken, The slaving principle for Stratanovich stochastic differential equations. Z. Phys. B 63, 493–504 (1986)ADSCrossRefGoogle Scholar
  41. 41.
    G. Schöner, H. Haken, A systematic elimination procedure for Itō stochastic differential equations and the adiabatic approximation. Z. Phys. B 68, 89–103 (1987)MathSciNetADSCrossRefzbMATHGoogle Scholar
  42. 42.
    R. Serra, M. Andretta, M. Compiani, G. Zanarini, Introduction to the Physics of Complex Systems (Pergamon Press, Oxford, 1986)zbMATHGoogle Scholar
  43. 43.
    N. Sri Namachchivaya, Equivalence of stochastic averaging and stochastic normal forms. J. Appl. Mech. 57, 1011–1017 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    N. Sri Namachchivaya, Y.K. Lin, Method of stochastic normal forms. Int. J. Nonlinear Mech. 26, 931–943 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    R.L. Stratanovich, Introduction to the Theory of Random Noise (Gordon and Breach, New York, 1963)Google Scholar
  46. 46.
    S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, New York, 2001)Google Scholar
  47. 47.
    P. Thomas, R. Grima, A.V. Straube, Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators. Phys. Rev. E 86, 041110 (2012)ADSCrossRefGoogle Scholar
  48. 48.
    N.G. van Kampen, Remarks on non-Markov processes. Braz. J. Phys. 28, 90–96 (1998)ADSGoogle Scholar
  49. 49.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 2007)Google Scholar
  50. 50.
    W. Wang, A.J. Roberts, Slow manifold and averaging for slow-fast stochastic differential system. J. Math. Anal. Appl. 398, 822–839 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    G. Wilemski, On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion. J. Stat. Phys. 14, 153–170 (1976)ADSCrossRefGoogle Scholar
  52. 52.
    S. Wright, Evolution in Mendelian populations. Genetics 16, 97–159 (1931)Google Scholar
  53. 53.
    R. Zwanzig, Ensemble method in the theory of irreversibility. J. Chem. Phys. 33, 1338–1341 (1960)MathSciNetADSCrossRefGoogle Scholar
  54. 54.
    R. Zwanzig, Memory effects in irreversible thermodynamics. Phys. Rev. 124, 983–992 (1961)ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Ecology and Evolutionary BiologyPrinceton UniversityPrincetonUSA

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