An oscillator driven by algebraically decorrelating noise

TL;DR

This paper studies a nonlinear oscillator driven by algebraically decorrelating Gaussian noise, showing that its scaled limit behaves like standard Brownian motion without memory, contrasting with the noise's long-range dependence.

Contribution

It demonstrates that a nonlinear oscillator driven by algebraically decaying noise converges to standard Brownian motion, not fractional Brownian motion, in the scaling limit.

Findings

The renormalized oscillator converges to diffusion driven by standard Brownian motion.

The noise's long-range dependence does not persist in the oscillator's scaling limit.

The proof uses the perturbed test function method in a fast-slow system framework.

Abstract

We consider a stochastically forced nonlinear oscillator driven by a stationary Gaussian noise that has an algebraically decaying covariance function. It is well known that such noise processes can be renormalized to converge to fractional Brownian motion, a process that has memory. In contrast, we show that the renormalized limit of the nonlinear oscillator driven by this noise converges to diffusion driven by standard (not fractional) Brownian motion, and thus retains no memory in the scaling limit. The proof is based on the study of a fast-slow system using the perturbed test function method.