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

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.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 reading the manuscript. We also thank P.E. Jabin for a discussion on Section 7. While working on the paper, Z. Pajor-Gyulai was partially supported by the NSF Grant Numbers 1309084 and DMS1101635. M. Salins was partially supported by the NSF Grant Number 1407615. The authors are also grateful for the anonymous referee for pointing out numerous typos and giving many suggestions, in particularly pointing us to several relevant papers that greatly improved the quality of the paper. (1309084 - NSF; DMS1101635 - NSF; 1407615 - NSF)Accepted manuscrip

Elements of Game Theory

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