Ergodic properties of functionals of Gaussian processes.

Area of Science:

• Statistical Physics
• Stochastic Processes
• Mathematical Physics

Background:

• Understanding the behavior of stochastic functionals is crucial in various scientific fields.
• Ergodicity is a fundamental concept in the study of stationary random walks.
• Probability density functions are key to analyzing random walk dynamics.

Purpose of the Study:

• To derive general results for the moments of positive stochastic functionals.
• To apply these results to Gaussian random walks, scaled Brownian motion, and fractional Brownian motion.
• To investigate ergodicity and universal properties of observables in these systems.

Main Methods:

• Derivation of moments using one- and two-time probability density functions.
• Analysis of occupation times (half-occupation and interval occupation).
• Application of infinite ergodic theory to identify universal properties.

Main Results:

• Exact analytic expressions for the first two moments of occupation times in Gaussian random walks.
• Computation of the ergodicity breaking parameter for scaled and fractional Brownian motion.
• Identification of universal scaling forms for occupation time probability densities.

Conclusions:

• The derived general results provide a powerful framework for analyzing stochastic functionals.
• The study confirms analytical predictions through numerical simulations.
• This research offers new insights into ergodicity and universal behaviors in complex stochastic systems.