Noise-Driven Return Statistics: Scaling and Truncation in Stochastic Storage Processes

In countless systems, subjected to variable forcing, a key question arises: how much time will a state variable spend away from a given threshold? When forcing is treated as a stochastic process, this can be addressed with first return time distributions. While many studies suggest exponential, double exponential or power laws as empirical forms, we contend that truncated power laws are natural candidates. To this end, we consider a minimal stochastic mass balance model and identify a parsimonious mechanism for the emergence of truncated power law return times. We derive boundary-independent scaling and truncation properties, which are consistent with numerical simulations, and discuss the implications and applicability of our findings.