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Abstract

What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals tau distributed as a power law similar to tau(-(1+alpha)); alpha > 0? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain 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.