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

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 ω 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 ω . 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 and numerical data.

Keywords

• Gaussian stochastic process
• power spectral density
• frequency–frequency correlation function