Communications in Mathematical Physics

, Volume 336, Issue 1, pp 131–170 | Cite as

Large Deviations and Gallavotti–Cohen Principle for Dissipative PDEs with Rough Noise

  • V. Jakšić
  • V. Nersesyan
  • C.-A. Pillet
  • A. Shirikyan


We study a class of dissipative PDEs perturbed by an unbounded kick force. Under some natural assumptions, the restrictions of solutions to integer times form a homogeneous Markov process. Assuming that the noise is rough with respect to the space variables and has a non-degenerate law, we prove that the system in question satisfies a large deviation principle (LDP) in τ-topology. Under some additional hypotheses, we establish a Gallavotti–Cohen type symmetry for the rate function of an entropy production functional and the strict positivity and finiteness of the mean entropy production rate in the stationary regime. The latter result is applicable to PDEs with strong nonlinear dissipation.


Entropy Production Burger Equation Stokes System Large Deviation Principle Entropy Production Rate 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BKL01.
    Bricmont J., Kupiainen A., Lefevere R.: Ergodicity of the 2d Navier–Stokes equations with random forcing. Commun. Math. Phys. 224(1), 65–81 (2001)CrossRefADSzbMATHMathSciNetGoogle Scholar
  2. BM05.
    Baiesi, M., Maes, C.: Enstrophy dissipation in two-dimensional turbulence. Phys. Rev. E (3) 72(5), 056314, 7 (2005)Google Scholar
  3. Bog98.
    Bogachev V.I.: Gaussian Measures, Mathematical Surveys and Monographs, vol. 62. American Mathematical Society, Providence (1998)Google Scholar
  4. Bor13.
    Boritchev A.: Estimates for solutions of a low-viscosity kick-forced generalized Burgers equation. Proc. R. Soc. Edinb. Sect. A 143(2), 253–268 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. BV92.
    Babin A.V., Vishik M.I.: Attractors of Evolution Equations. North-Holland Publishing, Amsterdam (1992)zbMATHGoogle Scholar
  6. Caz03.
    Cazenave T.: Semilinear Schrödinger Equations. New York University Courant Institute of Mathematical Sciences, New York (2003)zbMATHGoogle Scholar
  7. Doo48.
    Doob J.L.: Asymptotic properties of Markoff transition probabilities. Trans. Am. Math. Soc. 63, 393–421 (1948)zbMATHMathSciNetGoogle Scholar
  8. DS89.
    Deuschel J.-D., Stroock D.W.: Large Deviations. Academic Press, Boston (1989)zbMATHGoogle Scholar
  9. DZ96.
    Da Prato G., Zabczyk J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  10. DZ00.
    Dembo A., Zeitouni O.: Large Deviations Techniques and Applications. Springer, Berlin (2000)Google Scholar
  11. EH00.
    Eckmann J.-P., Hairer M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000)CrossRefADSzbMATHMathSciNetGoogle Scholar
  12. EPR99a.
    Eckmann J.-P., Pillet C.-A., Rey-Bellet L.: Entropy production in nonlinear, thermally driven Hamiltonian systems. J. Stat. Phys. 95(1–2), 305–331 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  13. EPR99b.
    Eckmann J.-P., Pillet C.-A., Rey-Bellet L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201(3), 657–697 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  14. ES94.
    Evans D.J., Searles D.J.: Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50, 1645–1648 (1994)CrossRefADSGoogle Scholar
  15. Gas04.
    Gaspard P.: Time-reversed dynamical entropy and irreversibility in Markovian random processes. J. Stat. Phys. 117(3–4), 599–615 (2004)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. GC95.
    Gallavotti G., Cohen E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80(5–6), 931–970 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  17. Gou07a.
    Gourcy M.: A large deviation principle for 2d stochastic Navier– Stokes equation. Stoch. Process. Appl. 117(7), 904–927 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  18. Gou07b.
    Gourcy M.: Large deviation principle of occupation measure for a stochastic Burgers equation. Ann. Inst. H. Poincaré Probab. Stat. 43(4), 375–408 (2007)CrossRefMathSciNetGoogle Scholar
  19. GS80.
    Gīhman Ĭ. Ī., Skorohod A.V.: The Theory of Stochastic Processes I. Springer, Berlin (1980)zbMATHGoogle Scholar
  20. GV96.
    Ginibre J., Velo G.: The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods. Phys. D 95(3–4), 191–228 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  21. JNPS12.
    Jakšić, V., Nersesyan, V., Pillet, C.-A., Shirikyan, A.: Large deviations from a stationary measure for a class of dissipative PDE’s with random kicks (2012, preprint) arXiv:1212.0527
  22. JPR11.
    Jakšić V., Pillet C.-A., Rey-Bellet L.: Entropic fluctuations in statistical mechanics: I. Classical dynamical systems. Nonlinearity 24(3), 699–763 (2011)CrossRefADSzbMATHMathSciNetGoogle Scholar
  23. Kru69.
    Kružkov S.N.: The Cauchy problem for certain classes of quasilinear parabolic equations. Mat. Zametki 6, 295–300 (1969)MathSciNetGoogle Scholar
  24. KS12.
    Kuksin S., Shirikyan A.: Mathematics of Two-Dimensional Turbulence. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  25. Kur98.
    Kurchan J.: Fluctuation theorem for stochastic dynamics. J. Phys. A 31(16), 3719–3729 (1998)CrossRefADSzbMATHMathSciNetGoogle Scholar
  26. LAv07.
    Lecomte V., Appert-Rolland C., van Wijland F.: Thermodynamic formalism for systems with Markov dynamics. J. Stat. Phys. 127(1), 51–106 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  27. LS99.
    Lebowitz J.L., Spohn H.: A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95(1–2), 333–365 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  28. Mae99.
    Maes C.: The fluctuation theorem as a Gibbs property. J. Stat. Phys. 95(1–2), 367–392 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  29. Mae04.
    Maes, C.: On the origin and the use of fluctuation relations for the entropy. Poincaré Seminar 2003, Prog. Math. Phys., vol. 38. Birkhäuser, Basel, pp. 145–191 (2004)Google Scholar
  30. Mét78.
    Métivier G.: Valeurs propres d’opérateurs définis par la restriction de systèmes variationnels à des sous-espaces. J. Math. Pures Appl. (9) 57(2), 133–156 (1978)zbMATHMathSciNetGoogle Scholar
  31. MN03.
    Maes C., Netočný K.: Time-reversal and entropy. J. Stat. Phys. 110(1–2), 269–310 (2003)CrossRefzbMATHGoogle Scholar
  32. MRV01.
    Maes, C., Redig, F., Verschuere, M.: From global to local fluctuation theorems. Mosc. Math. J. 1(3), 421–438, 471–472 (2001)Google Scholar
  33. MT93.
    Meyn S.P., Tweedie R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)CrossRefzbMATHGoogle Scholar
  34. Nov05.
    Novikov, D.: Hahn decomposition and Radon–Nikodym theorem with a parameter (2005). arXiv:math/0501215
  35. RM07.
    Rondoni L., Mejía-Monasterio C.: Fluctuations in nonequilibrium statistical mechanics: models, mathematical theory, physical mechanisms. Nonlinearity 20(10), R1–R37 (2007)CrossRefADSzbMATHGoogle Scholar
  36. RT02.
    Rey-Bellet L., Thomas L.E.: Fluctuations of the entropy production in anharmonic chains. Ann. Henri Poincaré 3(3), 483–502 (2002)CrossRefADSzbMATHMathSciNetGoogle Scholar
  37. Rue97.
    Ruelle D.: Entropy production in nonequilibrium statistical mechanics. Commun. Math. Phys. 189(2), 365–371 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  38. Rue99.
    Ruelle D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanic. J. Stat. Phys. 95(1–2), 393–468 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  39. Tay97.
    Taylor, M.E.: Partial Differential Equations. I–III. Springer, New York (1996–1997)Google Scholar
  40. Tem88.
    Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  41. Wei80.
    Weissler F.B.: Local existence and nonexistence for semilinear parabolic equations in L p. Indiana Univ. Math. J. 29(1), 79–102 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  42. Wu01.
    Wu L.: Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stoch. Process. Appl. 91(2), 205–238 (2001)CrossRefzbMATHGoogle Scholar
  43. Yag47.
    Yaglom A.M.: The ergodic principle for Markov processes with stationary distributions. Doklady Akad. Nauk SSSR (N.S.) 56, 347–349 (1947)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • V. Jakšić
    • 1
  • V. Nersesyan
    • 2
  • C.-A. Pillet
    • 3
    • 4
  • A. Shirikyan
    • 5
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Laboratoire de Matématiques, UMR CNRS 8100Université de Versailles-Saint-Quentin-en-YvelinesVersaillesFrance
  3. 3.Aix Marseille UniversitéMarseilleFrance
  4. 4.Univeristé de ToulonLa GardeFrance
  5. 5.Department of MathematicsUniversity of Cergy-PontoiseCergy-PontoiseFrance

Personalised recommendations