Beyond the Arcsine Law: Exact Two-Time Statistics of the Occupation Time in Jump Processes

TL;DR

This paper derives exact two-time occupation time statistics for one-dimensional jump processes, revealing universal features dependent only on jump distribution tails, advancing understanding of aging and correlations in complex stochastic systems.

Contribution

It provides the first exact characterization of two-time occupation statistics for generic jump processes, extending beyond renewal processes using generalized Wiener-Hopf methods.

Findings

Derived joint distribution of occupation time and position

Obtained the aged occupation-time law

Calculated the autocorrelation function

Abstract

Occupation times quantify how long a stochastic process remains in a region, and their single-time statistics are famously given by the arcsine law for Brownian and L\'evy processes. By contrast, two-time occupation statistics, which directly probe temporal correlations and aging, have resisted exact characterization beyond renewal processes. In this Letter we derive exact results for generic one-dimensional jump processes, a central framework for intermittent and discretely sampled dynamics. Using generalized Wiener-Hopf methods, we obtain the joint distribution of occupation time and position, the aged occupation-time law, and the autocorrelation function. In the continuous-time scaling limit, universal features emerge that depend only on the tail of the jump distribution, providing a starting point for exploring aging transport in complex environments.