Statistical properties of stochastic functionals under general resetting

Area of Science:

• Statistical Physics
• Stochastic Processes
• Random Walks

Background:

• Stochastic processes, particularly random walks, are fundamental in modeling diverse physical phenomena.
• Resetting mechanisms introduce unique dynamics, altering long-time behaviors and statistical properties.
• Understanding functionals of these processes is crucial for applications in physics and beyond.

Purpose of the Study:

• To derive the characteristic function of stochastic functionals for a reset random walk with general resetting time distributions.
• To analyze the long-time behavior and scaling properties of these functionals.
• To investigate the conditions for ergodicity and characterize the limiting distributions.

Main Methods:

• Derivation of characteristic functions for stochastic functionals.
• Analysis of long-time behavior and temporal scaling of moments.
• Investigation of probability density functions and ergodicity.
• Explicit examination of half-occupation time for Brownian and subdiffusive walks.
• Monte Carlo simulations for validation.

Main Results:

• Temporal scaling of moments is obtained for power-law resetting time distributions.
• Finite moments of resetting times lead to an ergodic phase where functional densities converge to a delta function.
• Ergodicity breaking parameter, moments, and limiting distributions are derived for power-law tails.
• Three distinct shapes of limiting distributions are characterized based on the resetting exponent.
• Simulations confirm excellent agreement with analytical results.

Conclusions:

• The study provides a comprehensive framework for analyzing reset random walks and their stochastic functionals.
• Power-law resetting distributions induce rich dynamical behaviors and ergodicity breaking.
• The findings offer insights into the statistical properties of systems with intermittent resetting mechanisms.