Diffusion with stochastic resetting at power-law times

Abstract

What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals $\tau$ distributed as a power-law $\sim \tau^{-(1+\alpha)};\alpha>0$? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain {\em exact} closed-form expressions for both static and dynamic quantities, while accounting for strong correlations implied by a power-law. Our results show that the resulting dynamics exhibits a spectrum of rich long-time behavior, from an ever-spreading spatial distribution for $\alpha < 1$, to one that is time independent for $\alpha > 1$. The dynamics has strong consequences on the time to reach a distant target for the first time; we specifically show that there exists an optimal $\alpha$ that minimizes the mean time to reach the target, thereby offering a step towards a viable strategy to locate targets in a crowded environment.