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arXiv · 2510.25859

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

Arthur Plaud·Olivier Bénichou
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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.

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Arthur Plaud, Olivier Bénichou. 2025-10-29. Beyond the Arcsine Law: Exact Two-Time Statistics of the Occupation Time in Jump Processes. https://doi.org/10.1103/ggwv-ckyj

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