Drift and Diffusion in Periodically Driven Renewal Processes

TL;DR

This paper develops a theoretical framework to analyze the drift and diffusion characteristics of periodically driven renewal processes, linking cumulant growth to the waiting time distribution and exploring implications for stochastic synchronization.

Contribution

It introduces a novel theory connecting periodic growth coefficients of cumulants to the waiting time distribution in periodically driven renewal processes.

Findings

Cumulants grow asymptotically periodically

Mean frequency and diffusion coefficient are derived

Framework quantifies stochastic synchronization

Abstract

We consider the drift and diffusion properties of periodically driven renewal processes. These processes are defined by a periodically time dependent waiting time distribution, which governs the interval between subsequent events. We show that the growth of the cumulants of the number of events is asymptotically periodic and develop a theory which relates these periodic growth coefficients to the waiting time distribution defining the periodic renewal process. The first two coefficients, which are the mean frequency and effective diffusion coefficient of the number of events are considered in greater detail. They may be used to quantify stochastic synchronization.