Advertisement

Annales Henri Poincaré

, Volume 16, Issue 9, pp 2005–2057 | Cite as

Nonequilibrium Markov Processes Conditioned on Large Deviations

  • Raphaël Chetrite
  • Hugo TouchetteEmail author
Article

Abstract

We consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob’s h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, often used with importance sampling to simulate rare events and giving rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned.

Keywords

Markov Process Canonical Ensemble Jump Process Large Deviation Principle Conditioned Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Doob J.L.: Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. Fr. 85, 431 (1957)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Doob J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York (1984)zbMATHGoogle Scholar
  3. 3.
    Rogers L.C.G., Williams D.: Diffusions, Markov Processes and Martingales. Cambridge University Press, Cambridge (2000)Google Scholar
  4. 4.
    Baudoin F.: Conditioned stochastic differential equations: Theory examples and application to finance. Stoch. Proc. Appl. 100, 109 (2002)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Gasbarra D., Sottinen T., Valkeila E.: Gaussian bridges. In: Benth, F.E., Nunno, G.D., Lindstrøm, T., Øksendal, B., Zhang, T. (eds.) Stochastic Analysis and Applications, Abel Symposia, vol. 2, pp. 361–382. Springer, Berlin (2007)Google Scholar
  6. 6.
    Sottinen T., Yazigi A.: Generalized Gaussian bridges. Stoch. Proc. Appl. 124, 3084 (2014)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Schrödinger, E.: Über die Umkehrung der Naturgesetze. Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl., 144–153 (1931)Google Scholar
  8. 8.
    Schrödinger E.: Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. Henri Poincaré 2, 269 (1932)zbMATHGoogle Scholar
  9. 9.
    Jamison B.: The Markov processes of Schrödinger. Z. Wahrsch. Verw. Gebiete 32, 323 (1975)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Zambrini J.C.: Stochastic mechanics according to Schrödinger. Phys. Rev. A 33, 1532 (1986)MathSciNetADSGoogle Scholar
  11. 11.
    Aebi R.: Schödinger Diffusion Processes. Birkhäuser, Basel (1996)Google Scholar
  12. 12.
    Darroch J.N., Seneta E.: On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probab. 2, 88 (1965)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Darroch J.N., Seneta E.: On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab. 4, 192 (1967)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Villemonais D.: Quasi-stationary distributions and population processes. Prob. Surveys 9, 340 (2012)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Collet P., Martínez S., Martín J.S.: Quasi-Stationary Distributions. Springer, New York (2013)zbMATHGoogle Scholar
  16. 16.
    van Doorn E.A., Pollett P.K.: Quasi-stationary distributions for discrete-state models. Eur. J. Oper. Res. 230, 1 (2013)Google Scholar
  17. 17.
    Vasicek O.A.: A conditional law of large numbers. Ann. Probab. 8, 142 (1980)MathSciNetGoogle Scholar
  18. 18.
    van Campenhout J.M., Cover T.M.: Maximum entropy and conditional probability. IEEE Trans. Inform. Theory 27, 483 (1981)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Cover T.M., Thomas J.A.: Elements of Information Theory. Wiley, New York (1991)zbMATHGoogle Scholar
  20. 20.
    Csiszar I.: Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab. 12, 768 (1984)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Dembo A., Zeitouni O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)zbMATHGoogle Scholar
  22. 22.
    Diaconis P., Freedman D.: A dozen de Finetti-style results in search of a theory. Ann. Inst. Henri Poincare B Probab. Stat. 23, 397 (1987)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Diaconis P., Freedman D.A.: Conditional limit theorems for exponential families and finite versions of de Finetti’s theorem. J. Theor. Probab. 1, 381 (1988)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Dembo A., Zeitouni O.: Refinements of the Gibbs conditioning principle. Probab. Theor. Relat. Fields 104, 1 (1996)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Csiszár I., Cover T.M., Choi B.-S.: Conditional limit theorems under Markov conditioning. IEEE Trans. Inform. Theory 33, 788 (1987)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Borkar V.S., Juneja S., Kherani A.A.: Peformance analysis conditioned on rare events: an adaptive simulation scheme. Commun. Inform. Syst. 3, 259 (2004)MathSciNetGoogle Scholar
  27. 27.
    Bucklew J.A.: Introduction to Rare Event Simulation. Springer, New York (2004)zbMATHGoogle Scholar
  28. 28.
    Feller W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York (1970)Google Scholar
  29. 29.
    Giardina C., Kurchan J., Peliti L.: Direct evaluation of large-deviation functions. Phys. Rev. Lett. 96, 120603 (2006)zbMATHADSGoogle Scholar
  30. 30.
    Lecomte V., Tailleur J.: A numerical approach to large deviations in continuous time. J. Stat. Mech. 2007, P03004 (2007)Google Scholar
  31. 31.
    Tailleur J., Lecomte V.: Simulation of large deviation functions using population dynamics. In: Marro, J., Garrido, P.L., Hurtado, P.I. (eds.) Modeling and Simulation of New Materials: Proceedings of Modeling and Simulation of New Materials, vol. 1091, pp. 212–219. AIP, Melville (2009)Google Scholar
  32. 32.
    Moral P.D.: Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York (2004)Google Scholar
  33. 33.
    Fleming W.H.: Exit probabilities and optimal stochastic control. Appl. Math. Optim. 4, 329 (1978)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Fleming W.H.: Logarithmic transformations and stochastic control. In: Fleming, W.H., Gorostiza, L.G. (eds.) Advances in Filtering and Optimal Stochastic Control, Lecture Notes in Control and Information Sciences, vol. 42, pp. 131–141. Springer, New York (1982)Google Scholar
  35. 35.
    Fleming W.H.: A stochastic control approach to some large deviations problems. In: Dolcetto, I.C., Fleming, W.H., Zolezzi, T. (eds.) Recent Mathematical Methods in Dynamic Programming, vol. 1119, pp. 52–66. Springer, New York (1985)Google Scholar
  36. 36.
    Fleming W.H., Sheu S.-J.: Stochastic variational formula for fundamental solutions of parabolic PDE. Appl. Math. Optim. 13, 193 (1985)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Kac M.: On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1 (1949)zbMATHGoogle Scholar
  38. 38.
    Stroock D.W., Varadhan R.S.: Multidimensional Diffusion Processes. Springer, New York (1979)zbMATHGoogle Scholar
  39. 39.
    Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)zbMATHGoogle Scholar
  40. 40.
    Fleming W.H., Soner H.M.: Controlled Markov Processes and Viscosity Solutions, Stochastic Modelling and Applied Probability, vol. 25. Springer, New York (2006)Google Scholar
  41. 41.
    Sheu S.-J.: Stochastic control and principal eigenvalue. Stochastics 11, 191 (1984)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Fleming W.H., Sheub S.J., Soner H.M.: A remark on the large deviations of an ergodic Markov process. Stochastics 22, 187 (1987)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Fleming W.H., Sheu S.-J.: Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential. Ann. Probab. 25, 1953 (1997)MathSciNetGoogle Scholar
  44. 44.
    Fleming W.H., McEneaney W.M.: Risk sensitive optimal control and differential games. In: Duncan, T.E., Pasik-Duncan, B. (eds.) Stochastic Theory and Adaptive Control, Lecture Notes in Control and Information Sciences, vol. 184, pp. 185–197. Springer, New York (1992)Google Scholar
  45. 45.
    Fleming W., McEneaney W.: Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim. 33, 1881 (1995)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Dupuis P., McEneaney W.: Risk-sensitive and robust escape criteria. SIAM J. Control Optim. 35, 2021 (1997)zbMATHMathSciNetGoogle Scholar
  47. 47.
    Nemoto T., Sasa S.-I.: Thermodynamic formula for the cumulant generating function of time-averaged current. Phys. Rev. E 84, 061113 (2011)ADSGoogle Scholar
  48. 48.
    Nemoto T., Sasa S.-I.: Variational formula for experimental determination of high-order correlations of current fluctuations in driven systems. Phys. Rev. E 83, 030105 (2011)ADSGoogle Scholar
  49. 49.
    Nemoto T., Sasa S.-I.: Computation of large deviation statistics via iterative measurement-and-feedback procedure. Phys. Rev. Lett. 112, 090602 (2014)ADSGoogle Scholar
  50. 50.
    Touchette H.: The large deviation approach to statistical mechanics. Phys. Rep. 478, 1 (2009)MathSciNetADSGoogle Scholar
  51. 51.
    Freidlin M.I., Wentzell A.D.: Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, vol. 260. Springer, New York (1984)Google Scholar
  52. 52.
    Gardiner C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, vol. 13, 2nd edn. Springer, New York (1985)Google Scholar
  53. 53.
    van Kampen N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)Google Scholar
  54. 54.
    Risken H.: The Fokker–Planck equation: Methods of solution and applications, 3rd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  55. 55.
    Evans R.M.L.: Rules for transition rates in nonequilibrium steady states. Phys. Rev. Lett. 92, 150601 (2004)zbMATHADSGoogle Scholar
  56. 56.
    Evans R.M.L.: Detailed balance has a counterpart in non-equilibrium steady states. J. Phys. A Math. Gen. 38, 293 (2005)zbMATHADSGoogle Scholar
  57. 57.
    Evans R.M.L.: Statistical physics of shear flow: a non-equilibrium problem. Contemp. Phys. 51, 413 (2010)ADSGoogle Scholar
  58. 58.
    Dellago C., Bolhuis P.G., Geissler P.L.: Transition path sampling. Adv. Chem. Phys. 123, 1 (2003)Google Scholar
  59. 59.
    Dellago C., Bolhuis P.G., Geissler P.L.: Transition path sampling methods. In: Ferrario, M., Ciccotti, G., Binder, K. (eds.) Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology, vol. 1, Lecture Notes in Physics, vol. 703, Springer, New York (2006)Google Scholar
  60. 60.
    Dellago C., Bolhuis P.: Transition path sampling and other advanced simulation techniques for rare events. In: Holm, C., Kremer, K. (eds.) Advanced Computer Simulation Approaches for Soft Matter Sciences III, Advances in Polymer Science, vol. 221, pp. 167–233. Springer, Berlin (2009)Google Scholar
  61. 61.
    Vanden-Eijnden E.: Transition path theory. In: Ferrario, M., Ciccotti, G., Binder, K. (eds.) Computer Simulations in Condensed Matter Systems From Materials to Chemical Biology, vol. 1, Lecture Notes in Physics, vol. 703, pp. 453–493. Springer, New York (2006)Google Scholar
  62. 62.
    Chandler D., Garrahan J.P.: Dynamics on the way to forming glass: bubbles in space-time. Annu. Rev. Chem. Phys. 61, 191 (2010)Google Scholar
  63. 63.
    Hedges L.O., Jack R.L., Garrahan J.P., Chandler D.: Dynamic order-disorder in atomistic models of structural glass formers. Science 323, 1309 (2009)ADSGoogle Scholar
  64. 64.
    Jack R.L., Sollich P.: Large deviations and ensembles of trajectories in stochastic models. Prog. Theor. Phys. Suppl. 184, 304 (2010)zbMATHADSGoogle Scholar
  65. 65.
    Lecomte V., Appert-Rolland C., van Wijland F.: Chaotic properties of systems with Markov dynamics. Phys. Rev. Lett. 95, 010601 (2005)ADSGoogle Scholar
  66. 66.
    Lecomte V., Appert-Rolland C., van Wijland F.: Thermodynamic formalism for systems with Markov dynamics. J. Stat. Phys. 127, 51 (2007)zbMATHMathSciNetADSGoogle Scholar
  67. 67.
    Garrahan J.P., Lesanovsky I.: Thermodynamics of quantum jump trajectories. Phys. Rev. Lett. 104, 160601 (2010)ADSGoogle Scholar
  68. 68.
    Garrahan J.P., Armour A.D., Lesanovsky I.: Quantum trajectory phase transitions in the micromaser. Phys. Rev. E 84, 021115 (2011)ADSGoogle Scholar
  69. 69.
    Ates C., Olmos B., Garrahan J.P., Lesanovsky I.: Dynamical phases and intermittency of the dissipative quantum ising model. Phys. Rev. A 85, 043620 (2012)ADSGoogle Scholar
  70. 70.
    Genway S., Garrahan J.P., Lesanovsky I., Armour A.D.: Phase transitions in trajectories of a superconducting single-electron transistor coupled to a resonator. Phys. Rev. E 85, 051122 (2012)ADSGoogle Scholar
  71. 71.
    Hickey J.M., Genway S., Lesanovsky I., Garrahan J.P.: Thermodynamics of quadrature trajectories in open quantum systems. Phys. Rev. A 86, 063824 (2012)ADSGoogle Scholar
  72. 72.
    Chetrite R., Touchette H.: Nonequilibrium microcanonical and canonical ensembles and their equivalence. Phys. Rev. Lett. 111, 120601 (2013)ADSGoogle Scholar
  73. 73.
    Karatzas I., Shreve S.: Methods of Mathematical Finance, Stochastic Modelling and Applied Probability, vol. 39. Springer, New York (1998)zbMATHGoogle Scholar
  74. 74.
    Berg H.C.: Random Walks in Biology. Princeton University Press, Princeton (1993)Google Scholar
  75. 75.
    Spohn H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)zbMATHGoogle Scholar
  76. 76.
    Kipnis C., Landim C.: Scaling Limits of Interacting Particle Systems, Grundlheren der mathematischen Wissenschaften, vol. 320. Springer, Berlin (1999)Google Scholar
  77. 77.
    Liggett T.M.: Interacting Particle Systems. Springer, New York (2004)Google Scholar
  78. 78.
    Derrida B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. 2007, P07023 (2007)MathSciNetGoogle Scholar
  79. 79.
    Bertini L., Sole A.D., Gabrielli D., Jona-Lasinio G., Landim C.: Stochastic interacting particle systems out of equilibrium. J. Stat. Mech. 2007, P07014 (2007)Google Scholar
  80. 80.
    Jacobs K.: Stochastic Processes for Physicists: Understanding Noisy Systems. Cambridge University Press, Cambridge (2010)Google Scholar
  81. 81.
    Krapivsky P.L., Redner S., Ben-Naim E.: A Kinetic View of Statistical Physics. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  82. 82.
    Nelson E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)zbMATHGoogle Scholar
  83. 83.
    Anderson W.J.: Continuous-Time Markov Chains: An Applications-Oriented Approach, Series in Statistics. Springer, New York (1991)Google Scholar
  84. 84.
    Chung K.L., Walsh J.B.: Markov Processes, Brownian Motion, and Time Symmetry, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  85. 85.
    Applebaum D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  86. 86.
    Sato K.: Lévy Processes and Infinite Divisibility, Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999)Google Scholar
  87. 87.
    Hörmander L.: Hypoelliptic second order differential equations. Acta. Math. 119, 147 (1967)zbMATHMathSciNetGoogle Scholar
  88. 88.
    Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Proc. Inter. Symp. Stoch. Diff. Equations, Kyoto, pp. 195–263. Wiley, New York (1978)Google Scholar
  89. 89.
    Lebowitz J.L., Spohn H.: A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333 (1999)zbMATHMathSciNetADSGoogle Scholar
  90. 90.
    Maes, C., Netočný, K.: Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states. Europhys. Lett. 82 (2008)Google Scholar
  91. 91.
    Baiesi M., Maes C., Wynants B.: Fluctuations and response of nonequilibrium states. Phys. Rev. Lett. 103, 010602 (2009)ADSGoogle Scholar
  92. 92.
    Jarzynski C.: Equilibrium free-energy differences from nonequilibrium measurements: a master-equation approach. Phys. Rev. E 56, 5018 (1997)ADSGoogle Scholar
  93. 93.
    Crooks G.E.: Nonequilibrium measurements of free energy differences for microscopically reversible markovian systems. J. Stat. Phys. 90, 1481 (1998)zbMATHMathSciNetADSGoogle Scholar
  94. 94.
    Sekimoto K.: Stochastic Energetics, Lect. Notes Phys., vol. 799. Springer, New York (2010)Google Scholar
  95. 95.
    Maes C., Netočný K., Wynants B.: Steady state statistics of driven diffusions. Phys. A 387, 2675 (2008)MathSciNetGoogle Scholar
  96. 96.
    Chernyak V., Chertkov M., Malinin S., Teodorescu R.: Non-equilibrium thermodynamics and topology of currents. J. Stat. Phys. 137, 109 (2009)zbMATHMathSciNetADSGoogle Scholar
  97. 97.
    Basu A., Bhattacharyya T., Borkar V.S.: A learning algorithm for risk-sensitive cost. Math. Oper. Res. 33, 880 (2008)zbMATHMathSciNetGoogle Scholar
  98. 98.
    den Hollander F.: Large Deviations. Fields Institute Monograph. Math. Soc., Providence (2000)Google Scholar
  99. 99.
    Ellis R.S.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)Google Scholar
  100. 100.
    Fortelle A.D.L.: Large deviation principle for Markov chains in continuous time. Probab. Inform. Trans. 37, 120 (2001)zbMATHGoogle Scholar
  101. 101.
    Bertini, L., Faggionato, A., Gabrielli, D.: Large deviations of the empirical flow for continuous time Markov chains (2012). arXiv:1210.2004
  102. 102.
    Roynette B., Yor M.: Penalising Brownian Paths. Springer, New York (2009)zbMATHGoogle Scholar
  103. 103.
    Lewis J.T., Pfister C.-E., Sullivan G.W.: The equivalence of ensembles for lattice systems: some examples and a counterexample. J. Stat. Phys. 77, 397 (1994)zbMATHMathSciNetADSGoogle Scholar
  104. 104.
    Lewis J.T., Pfister C.-E., Sullivan W.G.: Entropy, concentration of probability and conditional limit theorem. Markov Proc. Relat. Fields 1, 319 (1995)zbMATHMathSciNetGoogle Scholar
  105. 105.
    Touchette, H.: Equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels (2014). arXiv:1403.6608.
  106. 106.
    MacCluer C.R.: The many proofs and applications of Perron’s theorem. SIAM Rev. 42, 487 (2000)zbMATHMathSciNetADSGoogle Scholar
  107. 107.
    Krein M.G., Rutman M.A.: Linear operators leaving a cone in a Banach space. Am. Math. Soc. Transl. 26, 128 (1950)MathSciNetGoogle Scholar
  108. 108.
    Lecomte, V.: Thermodynamique des histoires et fluctuations hors d’équilibre. Ph.D. thesis, Université Paris VII (2007)Google Scholar
  109. 109.
    Chetrite R., Gupta S.: Two refreshing views of fluctuation theorems through kinematics elements and exponential martingale. J. Stat. Phys. 143, 543 (2011)zbMATHMathSciNetADSGoogle Scholar
  110. 110.
    Kunita H.: Absolute continuity of Markov processes and generators. Nagoya Math. J. 36, 1 (1969)zbMATHMathSciNetGoogle Scholar
  111. 111.
    Itô K., Watanabe S.: Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier 15, 13 (1965)zbMATHGoogle Scholar
  112. 112.
    Palmowski Z., Rolski T.: A technique for exponential change of measure for Markov processes. Bernoulli 8, 767 (2002)zbMATHMathSciNetGoogle Scholar
  113. 113.
    Diaconis P., Miclo L.: On characterizations of Metropolis type algorithms in continuous time. ALEA Lat. Am. J. Probab. Math. Stat. 6, 199 (2009)zbMATHMathSciNetGoogle Scholar
  114. 114.
    Meyer P.A., Zheng W.A.: Construction de processus de Nelson reversibles. In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités XIX 1983/84, Lecture Notes in Mathematics, vol. 1123, pp. 12–26. Springer, Berlin (1985)Google Scholar
  115. 115.
    Bernard W., Callen H.B.: Irreversible thermodynamics of nonlinear processes and noise in driven systems. Rev. Mod. Phys. 31, 1017 (1959)zbMATHMathSciNetADSGoogle Scholar
  116. 116.
    Callen H.B., Welton T.A.: Irreversibility and generalized noise. Phys. Rev. 83, 34 (1951)zbMATHMathSciNetADSGoogle Scholar
  117. 117.
    Kubo R.: The fluctuation–dissipation theorem. Rep. Prog. Phys. 29, 255 (1966)ADSGoogle Scholar
  118. 118.
    Lippiello E., Corberi F., Sarracino A., Zannetti M.: Nonlinear response and fluctuation–dissipation relations. Phys. Rev. E 78, 041120 (2008)ADSGoogle Scholar
  119. 119.
    Speck T., Seifert U.: Restoring a fluctuation–dissipation theorem in a nonequilibrium steady state. Europhys. Lett. 74, 391 (2006)ADSGoogle Scholar
  120. 120.
    Baiesi M., Maes C., Wynants B.: Nonequilibrium linear response for Markov dynamics I: jump processes and overdamped diffusions. J. Stat. Phys. 137, 1094 (2009)zbMATHMathSciNetADSGoogle Scholar
  121. 121.
    Baiesi M., Boksenbojm E., Maes C., Wynants B.: Nonequilibrium linear response for Markov dynamics II: Inertial dynamics. J. Stat. Phys. 139, 492 (2010)zbMATHMathSciNetADSGoogle Scholar
  122. 122.
    Seifert U., Speck T.: Fluctuation–dissipation theorem in nonequilibrium steady states. Europhys. Lett. 89, 10007 (2010)ADSGoogle Scholar
  123. 123.
    Chetrite R., Falkovich G., Gawedzki K.: Fluctuation relations in simple examples of non-equilibrium steady states. J. Stat. Mech. 2008, P08005 (2008)Google Scholar
  124. 124.
    Chetrite R., Gawedzki K.: Eulerian and Lagrangian pictures of non-equilibrium diffusions. J. Stat. Phys. 137, 890 (2009)zbMATHMathSciNetADSGoogle Scholar
  125. 125.
    Jarzynski C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690 (1997)ADSGoogle Scholar
  126. 126.
    Evans D.J., Cohen E.G.D., Morriss G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401 (1993)zbMATHADSGoogle Scholar
  127. 127.
    Gallavotti G., Cohen E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694 (1995)ADSGoogle Scholar
  128. 128.
    Gallavotti G., Cohen E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931 (1995)zbMATHMathSciNetADSGoogle Scholar
  129. 129.
    Dyson F.J.: A Brownian motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191 (1962)zbMATHMathSciNetADSGoogle Scholar
  130. 130.
    Grabiner D.J.: Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. Henri Poincare B Probab. Stat. 35, 177 (1999)zbMATHMathSciNetADSGoogle Scholar
  131. 131.
    O’Connell N.: Random matrices, non-colliding processes and queues. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds.) Séminaire de Probabilités XXXVI, Lecture Notes in Mathematics, vol. 1801, pp. 165–182. Springer, Berlin (2003)Google Scholar
  132. 132.
    Touchette H.: Ensemble equivalence for general many-body systems. Europhys. Lett. 96, 50010 (2011)ADSGoogle Scholar
  133. 133.
    Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  134. 134.
    Gingrich, T.R., Vaikuntanathan, S., Geissler, P.L.: Heterogeneity-induced large deviations in activity and (in some cases) entropy production (2014). arXiv:1406.3311
  135. 135.
    Merhav N., Kafri Y.: Bose–Einstein condensation in large deviations with applications to information systems. J. Stat. Mech. 2010, P02011 (2010)Google Scholar
  136. 136.
    Harris R.J., Popkov V., Schütz G.M.: Dynamics of instantaneous condensation in the ZRP conditioned on an atypical current. Entropy 15, 5065 (2013)ADSGoogle Scholar
  137. 137.
    Szavits-Nossan J., Evans M.R., Majumdar S.N.: Constraint-driven condensation in large fluctuations of linear statistics. Phys. Rev. Lett. 112, 020602 (2014)ADSGoogle Scholar
  138. 138.
    Baule A., Evans R.M.L.: Invariant quantities in shear flow. Phys. Rev. Lett. 101, 240601 (2008)ADSGoogle Scholar
  139. 139.
    Simha A., Evans R.M.L., Baule A.: Properties of a nonequilibrium heat bath. Phys. Rev. E 77, 031117 (2008)zbMATHADSGoogle Scholar
  140. 140.
    Baule A., Evans R.M.L.: Nonequilibrium statistical mechanics of shear flow: invariant quantities and current relations. J. Stat. Mech. 2010, P03030 (2010)MathSciNetGoogle Scholar
  141. 141.
    Popkov V., Schütz G.M., Simon D.: ASEP on a ring conditioned on enhanced flux. J. Stat. Mech. 2010, P10007 (2010)Google Scholar
  142. 142.
    Popkov V., Schütz G.: Transition probabilities and dynamic structure function in the ASEP conditioned on strong flux. J. Stat. Phys. 142, 627 (2011)zbMATHMathSciNetADSGoogle Scholar
  143. 143.
    Jack R.L., Sollich P.: Large deviations of the dynamical activity in the East model: Analysing structure in biased trajectories. J. Phys. A Math. Theor. 47, 015003 (2014)MathSciNetADSGoogle Scholar
  144. 144.
    Knežević M., Evans R.M.L.: Numerical comparison of a constrained path ensemble and a driven quasisteady state. Phys. Rev. E 89, 012132 (2014)ADSGoogle Scholar
  145. 145.
    Chen J., Li H., Jian S.: Some limit theorems for absorbing Markov processes. J. Phys. A Math. Theor. 45, 345003 (2012)MathSciNetGoogle Scholar
  146. 146.
    Chen J., Deng X.: Large deviations and related problems for absorbing Markov chains. Stoch. Proc. Appl. 123, 2398 (2013)zbMATHMathSciNetGoogle Scholar
  147. 147.
    Bauer, M., Cornu, F.: Affinity and fluctuations in a mesoscopic noria (2014). arXiv:1402.2422
  148. 148.
    Miller H.D.: A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Stat. 32, 1260 (1961)zbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Laboratoire J. A. Dieudonné, UMR CNRS 7351Universcontactité de Nice Sophia AntipolisNiceFrance
  2. 2.National Institute for Theoretical Physics (NITheP)StellenboschSouth Africa
  3. 3.Institute of Theoretical PhysicsStellenbosch UniversityStellenboschSouth Africa

Personalised recommendations