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

We investigate the stochastic behavior of the single-trajectory spectral density S(omega,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 w and the observation time T. We evaluate in particular the variance and the frequency-frequency correlation of S(omega,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 and numerical data.

Automated summary

This study investigates the stochastic behavior of the single-trajectory spectral density S(omega,T) of various Gaussian stochastic processes, such as Brownian motion, Ornstein-Uhlenbeck process, Brownian gyrator model, and fractional Brownian motion, in terms of frequency w and observation time T. The variance and frequency-frequency correlation of S(omega,T) are evaluated for different values of omega. The results demonstrate that these properties exhibit different behaviors for different physical cases, thus serving as a sensitive probe to distinguish between different types of random motion. These findings are of great importance in the analysis of experimental and numerical data.