Anomalous diffusion in a random nonlinear oscillator due to high frequencies of the noise

TL;DR

This paper investigates how a nonlinear oscillator's long-term behavior is affected by high-frequency noise, revealing power-law growth in observables and dependence on potential and noise spectrum.

Contribution

It provides a detailed analysis of anomalous diffusion caused by high-frequency noise in nonlinear oscillators, including scaling exponents and effects of dissipation.

Findings

Amplitude, velocity, and energy grow as power-laws over time

Scaling exponents depend on potential and noise spectrum

Results extend to dissipative systems with additive noise

Abstract

We study the long time behaviour of a nonlinear oscillator subject to a random multiplicative noise with a spectral density (or power-spectrum) that decays as a power law at high frequencies. When the dissipation is negligible, physical observables, such as the amplitude, the velocity and the energy of the oscillator grow as power-laws with time. We calculate the associated scaling exponents and we show that their values depend on the asymptotic behaviour of the external potential and on the high frequencies of the noise. Our results are generalized to include dissipative effects and additive noise.