On Dynamical Systems Perturbed by a Null-Recurrent Fast Motion: The Continuous Coefficient Case with Independent Driving Noises
- 83 Downloads
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.
KeywordsAveraging 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.
- 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
- 7.Khasminskii, R.Z.: On averaging principle for Ito stochastic differential equations. Kybernetika, Chekhoslovakia 4(3), 260–279 (1968). (in Russian)Google Scholar
- 11.Papanicolaou, G.C., Stroock, D., Varadhan, S.R.S.: Martingale approach to some limit theorems, Duke Turbulence Conference (1977)Google Scholar
- 12.Pavliotis, G.A., Stuart, A.: Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics 53. Springer, New York (2008)Google Scholar
- 13.Skorokhod, A.: Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monograhps, 78. AMS, Providence, RI (1989)Google Scholar