Spectral density of an individual trajectory of an arbitrary Gaussian stochastic process

by Gleb Oshanin (LPTMC - Université Pierre et Marie Curie)

Tuesday 16 Apr 2024, 13:00 → 14:00 Europe/Rome

Aula 6 (Dip. di Fisica - Edificio E. Fermi)

In this talk I will focus on the behavior of a particular random functional - the spectral density S(f,T) (with f being the frequency and T - the observation time) of an individual trajectory of an arbitrary stochastic centered Gaussian process. I will first recall the textbook definition based on the covariance function of the process, and show on several examples how diverse its functional form can be depending on a spread and a precise definition of the process. Then, I will specify the limitations of the standard definition and will go beyond it by considering the “noise-to-signal” ratio - the ratio of the standard deviation of S(f,T) and its mean value. Next, I will prove a simple but crucial double-sided inequality obeyed by the noise-to-signal ratio for any Gaussian process, any f and any T, and eventually will derive the full probability density function of S(f,T), under most general conditions. Lastly, for several Gaussian processes (as exemplified by Brownian motion, Ornstein-Uhlenbeck process, Brownian gyrator and fractional Brownian motion) I will discuss the behavior of the frequency-frequency correlations of such random variables and will demonstrate that they may be used as a robust property permitting to distinguish between normal and anomalous diffusion.