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Exponential mixing under controllability conditions for sdes driven by a degenerate Poisson noise

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  • Nersesyan, Vahagn
  • Raquépas, Renaud

Abstract

We prove existence and uniqueness of the invariant measure and exponential mixing in the total-variation norm for a class of stochastic differential equations driven by degenerate compound Poisson processes. In addition to mild assumptions on the distribution of the jumps for the driving process, the hypotheses for our main result are that the corresponding control system is dissipative, approximately controllable and solidly controllable. The solid controllability assumption is weaker than the well-known parabolic Hörmander condition and is only required from a single point to which the system is approximately controllable. Our analysis applies to Galerkin projections of stochastically forced parabolic partial differential equations with asymptotically polynomial nonlinearities and to networks of quasi-harmonic oscillators connected to different Poissonian baths.

Suggested Citation

  • Nersesyan, Vahagn & Raquépas, Renaud, 2021. "Exponential mixing under controllability conditions for sdes driven by a degenerate Poisson noise," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 26-55.
    • Handle: RePEc:eee:spapps:v:138:y:2021:i:c:p:26-55
    DOI: 10.1016/j.spa.2021.04.001
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    References listed on IDEAS

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    1. Morgado, Welles A.M. & Guerreiro, Thiago, 2012. "A study on the action of non-Gaussian noise on a Brownian particle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(15), pages 3816-3827.
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