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Statistical properties of stochastic functionals under general resetting.

Vicenç Méndez1, Rosa Flaquer-Galmés1

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This study analyzes stochastic functionals of random walks with resetting. We found that power-law resetting times lead to distinct behaviors, including an ergodic phase and specific distributions for functionals like half-occupation time.

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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.