Drift and diffusion in periodically driven renewal processes (Q2502369)

Let \(t_0\leq t_1\leq t_2\leq\dots\) be a sequence of renewal times, whose conditional distributions has a density \(w\), i.e.\ \(P(t_{i+1}\in[t+\tau, t+\tau+d\tau)\mid t_i=t)=w(\tau,t)\,d\tau\). It is assumed that \(w(\tau,t)=w(\tau,t+T)\) for some period \(T>0\). The authors study probabilistic properties of the number of times \(t_i\) on the interval \([t_0,t]\), \(N_{t_0,t}=\sum_{i=0}^\infty\mathbf{1}\) \((t_i\leq t)\). For example, the mean frequency and effective diffusion coefficient are evaluated with help of the generating function of \(N\). The authors also obtain a master equation for the probabilities \(p(k)=P(N_{t_0,t}=k)\) and embed the discrete dynamics into a continuous one. Finally, the theoretical and numerical results are compared for a simple model with \(w(\tau,t)=\theta\gamma(\tau-T(t))\exp(-\gamma(\tau-T(t)))\), \(T(t)\) being a stepwise periodic function with two values.