Journal of Theoretical Probability

, Volume 29, Issue 3, pp 1083–1099 | Cite as

On Dynamical Systems Perturbed by a Null-Recurrent Fast Motion: The Continuous Coefficient Case with Independent Driving Noises

  • Zsolt Pajor-Gyulai
  • Michael Salins


An ordinary differential equation perturbed by a null-recurrent diffusion will be considered in the case where the averaging type perturbation is strong only when a fast motion is close to the origin. The normal deviations of these solutions from the averaged motion are studied, and a central limit type theorem is proved. The limit process satisfies a linear equation driven by a Brownian motion time changed by the local time of the fast motion.


Averaging Null-recurrent fast motion Brownian local time  Normal deviations 

Mathematics Subject Classification (2010)

60H10 60J60 60F05 



The authors are grateful to D. Dolgopyat for introducing them to the problem and to L. Koralov and D. Dolgopyat for their helpful suggestions during invaluable discussions and for proofreading the manuscript. While working on the paper, Z. Pajor-Gyulai was partially supported by NSF Grants Number 1309084 and DMS1101635. M. Salins was partially supported by NSF Grant Number 1407615.


  1. 1.
    Ben Arous, G., Černý, J.: Scaling limit for trap models on \(\mathbb{Z}^d\). Ann. Probab. 35(6), 2356–2384 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Freidlin, M., Wentzell, A.: Diffusion processes on an open book and the averaging principle. Stoch. Process. Their Appl. 113, 101–126 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Freidlin, M., Wentzell, A.: Random Perturbations of Dynamical Systems. Springer, New York (2012)CrossRefMATHGoogle Scholar
  4. 4.
    Gikhman, I., Skorokhod, A.: Stochastic Differential Equations and their Applications. Springer, New York (1972)MATHGoogle Scholar
  5. 5.
    Hairer, M., Koralov, L., Pajor-Gyulai, Z.: Averaging and Homogenization in cellular flows: An exact description of the phase transition, Submitted to Annales d’Institut Henri Poincare (2014)Google Scholar
  6. 6.
    Hairer, M., Manson, C.: Periodic homogenization with an interface: The multidimensional case. Ann Probab 39(2), 648–682 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Khasminskii, R.Z.: On averaging principle for Ito stochastic differential equations. Kybernetika, Chekhoslovakia 4(3), 260–279 (1968). (in Russian)Google Scholar
  8. 8.
    Khasminskii, R.Z., Krylov, N.: On averaging principle for diffusion processes with null-recurrent fast component. Stoch Process Their Appl 93, 229–240 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kifer, Y.: Some Recent Advances in Averaging, Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  10. 10.
    Khasminskii, R.Z., Yin, G.: On averaging principlses: an asymptotic expansion approach. SIAM J Math Anal 35(6), 1534–1560 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Papanicolaou, G.C., Stroock, D., Varadhan, S.R.S.: Martingale approach to some limit theorems, Duke Turbulence Conference (1977)Google Scholar
  12. 12.
    Pavliotis, G.A., Stuart, A.: Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics 53. Springer, New York (2008)Google Scholar
  13. 13.
    Skorokhod, A.: Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monograhps, 78. AMS, Providence, RI (1989)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Maryland, College ParkCollege ParkUSA

Personalised recommendations