Statistical properties of stochastic functionals under general resetting

TL;DR

This paper investigates the statistical properties of stochastic functionals of random walks with general resetting, deriving characteristic functions, analyzing long-time behavior, and exploring ergodicity and distribution shapes under various resetting time distributions.

Contribution

It provides a general analytical framework for stochastic functionals under arbitrary resetting distributions, including power-law tails, and characterizes the resulting ergodic and non-ergodic behaviors.

Findings

Derived the characteristic function of stochastic functionals with resetting.

Identified conditions for ergodic and non-ergodic phases based on resetting distribution.

Characterized the limiting distribution shapes as a function of the resetting tail exponent.

Abstract

We derive the characteristic function of stochastic functionals of a random walk whose position is reset to the origin at random times drawn from a general probability distribution. We analyze the long-time behavior and obtain the temporal scaling of the first two moments of any stochastic functional of the random walk when the resetting time distribution exhibits a power-law tail. When the resetting times PDF has finite moments, the probability density of any functional converges to a delta function centered at its mean, indicating an ergodic phase. We explicitly examine the case of the half-occupation time and derive the ergodicity breaking parameter, the first two moments, and the limiting distribution when the resetting time distribution follows a power-law tail, for both Brownian and subdiffusive random walks. We characterize the three different shapes of the limiting distribution as a function of the exponent of the resetting distribution. Our theoretical findings are supported by Monte Carlo simulations, which show excellent agreement with the analytical results.