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On dynamical systems perturbed by a null-recurrent motion: The general case

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  • Pajor-Gyulai, Zs.
  • Salins, M.

Abstract

We consider a perturbed ordinary differential equation where the perturbation is only significant when a one-dimensional null recurrent diffusion is close to zero. We investigate the first order correction to the unperturbed system and prove a central limit theorem type result, i.e., that the normalized deviation process converges in law in the space of continuous functions to a limit process which we identify. We show that this limit process has a component which only moves when the limit of the null-recurrent fast motion equals zero. The set of these times forms a zero-measure Cantor set and therefore the limiting process cannot be described by a standard SDE. We characterize this process by its infinitesimal generator (with appropriate boundary conditions) and we also characterize the process as the weak solution of an SDE that depends on the local time of the fast motion process. We also investigate the long time behavior of such a system when the unperturbed motion is trivial. In this case, we show that the long-time limit is constant on a set of full Lebesgue measure with probability 1, but it has nontrivial drift and diffusion components that move only when the fast motion equals zero.

Suggested Citation

  • Pajor-Gyulai, Zs. & Salins, M., 2017. "On dynamical systems perturbed by a null-recurrent motion: The general case," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1960-1997.
  • Handle: RePEc:eee:spapps:v:127:y:2017:i:6:p:1960-1997
    DOI: 10.1016/j.spa.2016.09.009
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    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    3. Khasminskii, R. & Krylov, N., 2001. "On averaging principle for diffusion processes with null-recurrent fast component," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 229-240, June.
    4. Blei, Stefan & Engelbert, Hans-Jürgen, 2013. "One-dimensional stochastic differential equations with generalized and singular drift," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4337-4372.
    5. Krylov, N. V., 2004. "On weak uniqueness for some diffusions with discontinuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 113(1), pages 37-64, September.
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