Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes

TL;DR

This paper studies the spectral density of various Gaussian processes, analyzing their variance and frequency correlations to distinguish different types of stochastic motion, aiding experimental data interpretation.

Contribution

It provides a detailed analysis of the variance and frequency correlations of spectral densities for multiple Gaussian processes, highlighting their potential to differentiate stochastic behaviors.

Findings

Spectral density variance varies across processes.

Frequency correlations differ significantly among models.

Results can improve analysis of experimental stochastic data.

Abstract

We investigate the stochastic behavior of the single-trajectory spectral density $S(\omega,\mathcal{T})$ of several Gaussian stochastic processes, i.e., Brownian motion, the Ornstein-Uhlenbeck process, the Brownian gyrator model and fractional Brownian motion, as a function of the frequency $\omega$ and the observation time $\mathcal{T}$. We evaluate in particular the variance and the frequency-frequency correlation of $S(\omega,\mathcal{T})$ for different values of $\omega$. We show that these properties exhibit different behaviors for different physical cases and can therefore be used as a sensitive probe discriminating between different kinds of random motion. These results may prove quite useful in the analysis of experimental data.