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

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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 (AB) 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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Correspondence to Renaud Raquépas.

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Communicated by Christian Maes.

The author would like to thank Armen Shirikyan for introduction to these questions and crucial suggestions for this particular application, Noé Cuneo and Vojkan Jakšić for informative discussions, as well as the Département de mathématiques at Université Cergy–Pontoise, where part of this research was conducted, for its hospitality. The research of the author was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec–Nature et technologies (FRQNT) and the NonStops project of the Agence nationale de la recherche (ANR-17-CE40-0006-02).

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Raquépas, R. A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks. Ann. Henri Poincaré 20, 605–629 (2019). https://doi.org/10.1007/s00023-018-0740-0

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  • DOI: https://doi.org/10.1007/s00023-018-0740-0

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