On dynamical systems perturbed by a null-recurrent fast motion: The continuous coefficient case with independent driving noises

TL;DR

This paper investigates the behavior of ordinary differential equations influenced by null-recurrent diffusions, establishing a central limit theorem where the limit process involves a Brownian motion modified by local time.

Contribution

It introduces a novel analysis of ODEs with null-recurrent perturbations, deriving a central limit theorem with a limit process driven by a time-changed Brownian motion.

Findings

Proves a central limit theorem for solutions near averaged motion.

Shows the limit process is driven by a Brownian motion with local time modification.

Provides insights into the impact of null-recurrent diffusions on differential equations.

Abstract

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.