Nonstandard diffusion under Markovian resetting in bounded domains

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results, we find that when the waiting time probability has first and second finite moments, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset point to the boundary, or (iii) always beneficial. Instead, when the waiting time probability has the first or the two first moments diverging we find that resetting is always beneficial. Finally, we have also found that the optimal strategy to exit the domain depends on the reset rate. For low reset rates, walkers with exponential waiting times are found to be optimal, while for high reset rate, anomalous waiting times optimize the search process.

Figure 1

Parameter space diagram for the different existence regions of optimal reset rate.

Figure 2

Plot for the MFET vs $r\tau$ for different values of $x_0/L$ when the waiting time PDF is a Pareto PDF with $\alpha =3/2$. Solid curves correspond to theoretical result in Eq. (30) while symbols are the numerical simulations. $L=1$ and $\sigma =0.1x_0$. When $r\to 0$ the MFET tends to a finite value given by taking this limit to Eq. (41), that is $T_0\left(x_0\right)=x_0(L-x_0)\tau /{\sigma }^2(\alpha -1)$.

Figure 3

Plot for the MFET vs $r\tau$ for different values of $x_0/L$ when the waiting time PDF is a Mittag-Leffler function with $\gamma =0.7$. Solid curves correspond to theoretical result in Eq. (43) while symbols are the numerical simulations. $L=1$ and $\sigma =0.1x_0$. In the limit $r\to 0$ from Eq. (44) $T_r\left(x_0\right)\sim r^{-1+\gamma }$, i.e., the MFET diverges, although it cannot be observed in the figure because the lowest value shown is ${10}^{-8}$.

Figure 4

Plot for the optimal reset rate $r_{\mathrm{opt}}$ vs. the anomalous exponent $\gamma$ (for $\sigma ={10}^{-3}$ and $L=1$). The solid curves correspond to the analytical result given by Eq. (45).

Figure 5

Plot for $T_r(x_0,\gamma )$ vs $r\tau$. Symbols correspond to numerical simulations and solid curves are the theoretical predictions given by Eq. (43). $L=1$, $x_0=0.1$, and $\sigma =0.01$.