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

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 Lévy 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.

Figures (3)

• Figure 1: A discrete-time random walk $X_n$ (i.e., a jump process) starting at $X_0=0$. The first three excursions—time intervals between consecutive sign changes of $X_n$—have durations $t_i$. Segments with $X_n<0$ are shown in blue and those with $X_n>0$ in red. Each excursion ends with a jump that crosses the origin; the nonzero landing position defines the overshoot (red arrows). Large excursions typically end with large overshoots, which restart the next excursion farther from $0$ and tend to lengthen it. These overshoot-induced dependencies couple successive $t_i$, so the sequence of excursions is not renewal (durations are not i.i.d.).
• Figure 2: The regular part $f_{\text{reg}}(s,r)$ of the limiting distribution of the aged occupation time, obtained by numerically solving \ref{['eq:integral_aged']}. On the left, the distribution is shown in the Cauchy case $\alpha =1$ for $3$ different values of the aging ratio $r$. On the right this time, the aging ratio is fixed to $1$ but $3$ different universality classes are presented. Agreement with numerical simulations (triangles) is excellent.
• Figure 3: The limiting distribution of the rescaled forward recurrence time $F_n/n$, for three values of $\alpha$. For large $u$, $f_{\text{FRT}}(u)\sim \frac{2}{\pi \alpha}\sin\!\bigl(\frac{\pi \alpha}{4}\bigr)\,u^{-3/2}$, consistent with the universal Sparre–Andersen prediction. The small-$u$ behavior diverges for $\alpha \geq 1$ and remains finite for $\alpha < 1$. Agreement with numerical simulations (triangles) is excellent.