Diffusion with stochastic resetting at power-law times

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 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.

Figure 1

Typical space-time trajectories (red lines), with black lines marking resetting events: Resetting location $x_0=0$, diffusion constant $D=0.5, {\tau }_0=1.0$.

Figure 2

(a), (b) Data collapse of exact spatial distribution for $\alpha <1$ for different times, following Eq. (5). (c) Time-independent distribution for $\alpha >1$, Eq. (8). Resetting location $x_0=0$, diffusion constant $D=0.5, {\tau }_0=1.0$.

Figure 3

$\overline{T}\left(\alpha \right)$ versus $\alpha$, showing the existence of a minimum.