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Positive Harris recurrence for degenerate diffusions with internal variables and randomly perturbed time-periodic input

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  • Holbach, Simon

Abstract

We consider a multidimensional time-homogeneous dynamical system and add a randomly perturbed time-dependent deterministic signal to some of its components, giving rise to a high-dimensional system of stochastic differential equations, which is driven by possibly very low-dimensional noise. Equations of this type commonly occur in biology when modeling neurons or in statistical mechanics for certain Hamiltonian systems. We provide verifiable conditions on the original deterministic dynamical system under which the solution to the respective stochastic system features a point in the interior of its state space, which can be proved to be attainable by deterministic control arguments, and at which a local Hörmander condition holds. Together with a Lyapunov condition, it follows that the corresponding process is positive Harris recurrent.

Suggested Citation

  • Holbach, Simon, 2020. "Positive Harris recurrence for degenerate diffusions with internal variables and randomly perturbed time-periodic input," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6965-7003.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:11:p:6965-7003
    DOI: 10.1016/j.spa.2020.07.005
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    1. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    2. Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
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    Cited by:

    1. Reinhard Höpfner, 2021. "Polynomials under Ornstein–Uhlenbeck noise and an application to inference in stochastic Hodgkin–Huxley systems," Statistical Inference for Stochastic Processes, Springer, vol. 24(1), pages 35-59, April.

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