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

TL;DR

This paper studies the behavior of perturbed differential equations influenced by a null-recurrent diffusion, revealing a complex limit process characterized by local time and a zero-measure set, extending classical stochastic analysis.

Contribution

It introduces a novel analysis of the first order correction and limit process for systems perturbed by null-recurrent motions, including characterization via generator and SDE with local time.

Findings

The limit process involves a zero-measure Cantor set of times.

The process is characterized by its infinitesimal generator and local time dependence.

Long-time behavior shows a constant limit with nontrivial local-time-driven components.

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.