Transition behavior of the waiting time distribution in a stochastic model with the internal state

It has been noticed that when the waiting time distribution exhibits a transition from an intermediate time power-law decay to a long-time exponential decay in the continuous time random walk model, a transition from anomalous diffusion to normal diffusion can be observed at the population level. However, the mechanism behind the transition of waiting time distribution is rarely studied. In this paper, we provide one possible mechanism to explain the origin of such a transition. A stochastic model terminated by a state-dependent Poisson clock is studied by a formal asymptotic analysis for the time evolutionary equation of its probability density function (PDF). The waiting time behavior under a more relaxed setting can be rigorously characterized by probability tools. Both approaches show the transition phenomenon of the waiting time T, which is complemented by particle simulations to shed light on the transition time scale. Our results indicate that small drift relative to noise in the state equation and a stiff response in the Poisson rate are crucial to the transitional phenomena.