Ergodic properties of functionals of Gaussian processes

Abstract

We derive the first two moments of generic positive stochastic functionals in terms of the one- and two-time probability density functions of the underlying random walk, and we prove ergodicity of observables in stationary random walks. These general results are applied to the half-occupation time and the occupation time in an interval of a Gaussian random walk, for which we obtain exact analytic expressions for the first two moments. We then extend the analysis to scaled Brownian motion and fractional Brownian motion, computing the ergodicity breaking parameter and establishing a simple scaling form for the probability densities of occupation times. Within the framework of infinite ergodic theory, we further identify universal properties of positive observables. All analytical predictions are fully confirmed by numerical simulations.

References in corpus (10)

• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
• Spectral content of a single non-Brownian trajectory
• On distributions of functionals of anomalous diffusion paths
• Generalized arcsine laws for fractional Brownian motion
• Aging Scaled Brownian Motion
• Functionals of the Brownian motion, localization and metric graphs
• Statistical Properties of Functionals of the Paths of a Particle Diffusing in a One-Dimensional Random Potential
• Anomalous diffusion and long-range memory in the scaled voter model
• Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls