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Annales Henri Poincaré

, Volume 20, Issue 2, pp 605–629 | Cite as

A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks

  • Renaud RaquépasEmail author
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Abstract

We show that elements of control theory, together with an application of Harris’ ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the form
$$\begin{aligned} \mathrm{d}X_{t} = A X_{t} \mathrm{d}t + F(X_{t}) \mathrm{d}t + B \mathrm{d}W_{t} \end{aligned}$$
in \(\mathbf {R}^d\), where A encodes the harmonic part of the force and \(-F\) corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: A is dissipative, the pair (A, B) satisfies the Kalman condition, F grows sufficiently slowly at infinity (depending on the dimension d), and the vector fields in the equation of motion satisfy the weak Hörmander condition in at least one point of the phase space.
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References

  1. 1.
    Agrachev, A.A., Kuksin, S., Sarychev, A.V., Shirikyan, A.: On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier–Stokes equations. Annales de l’Institut Henri Poincaré (B) Probability and Statistics 43(4), 399–415 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agrachev, A.A., Sarychev, A.V.: Navier–Stokes equations: controllability by means of low modes forcing. J. Math. Fluid Mech. 7(1), 108–152 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogachev, V.I.: Gaussian Measures, Number 62. American Mathematical Society, Providence (1998)Google Scholar
  4. 4.
    Carmona, P.: Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths. Stoch. Process. Appl. 117(8), 1076–1092 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Coron, J.-M.: Control and Nonlinearity, Volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2007)Google Scholar
  6. 6.
    Cuneo, N., Eckmann, J.-P., Hairer, M., Rey-Bellet, L.: Non-equilibrium steady states for networks of oscillators. Electron. J. Probab. 23, 1–28 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Entropy production in nonlinear, thermally driven hamiltonian systems. J. Stat. Phys. 95(1), 305–331 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ford, G.W., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6(4), 504–515 (1965)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hairer, M., Mattingly, J.C.: Yet another look at Harris’ ergodic theorem for Markov chains. In: Seminar on Stochastic Analysis, Random Fields and Applications VI, volume 63, pp. 109–117. Springer, Berlin (2011)Google Scholar
  12. 12.
    Harris, T.E.: The existence of stationary measures for certain Markov processes. Proc. Third Berkeley Symp. Math. Stat. Probab. 2, 113–124 (1956)MathSciNetGoogle Scholar
  13. 13.
    Jakšić, V., Pillet, C.-A., Shirikyan, A.: Entropic fluctuations in thermally driven harmonic networks. J. Stat. Phys. 166(3), 926–1015 (2017)ADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer, Berlin (2012)Google Scholar
  15. 15.
    Rey-Bellet, L., Thomas, L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225(2), 305–329 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shirikyan, A.: Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations. J. Funct. Anal. 249, 284–306 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shirikyan, A.: Controllability implies mixing I. Convergence in the total variation metric. Uspekhi Matematicheskikh Nauk 72(5), 165–180 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tropper, M.M.: Ergodic and quasideterministic properties of finite-dimensional stochastic systems. J. Stat. Phys. 17(6), 491–509 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Univ. Grenoble AlpesCNRS, Institut FourierGrenobleFrance
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada

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