First-passage time of a Brownian searcher with stochastic resetting to random positions

We study the effect of a resetting point randomly distributed around the origin on the mean first-passage time of a Brownian searcher moving in one dimension. We compare the search efficiency with that corresponding to reset to the origin and find that the mean first-passage time of the latter can be larger or smaller than the distributed case, depending on whether the resetting points are symmetrically or asymmetrically distributed. In particular, we prove the existence of an optimal reset rate that minimizes the mean first-passage time for distributed resetting to a finite interval if the target is located outside this interval. When the target position belongs to the resetting interval or it is infinite then no optimal reset rate exists, but there is an optimal resetting interval width or resetting characteristic scale which minimizes the mean first-passage time. We also show that the first-passage density averaged over the resetting points depends on its first moment only. As a consequence, there is an equivalent point such that the first-passage problem with resetting to that point is statistically equivalent to the case of distributed resetting. We end our study by analyzing the fluctuations of the first-passage times for these cases. All our analytical results are verified through numerical simulations.

Figure 1

AMFPT for uniformly distributed reset points PDF in an interval $[-L,L]$ for $x_T\notin \mathcal{D}$. The solid line represents Eq. (23). The data points are obtained from numerical simulations. In blue (triangle) we present the Evans and Majumdar result, Eq. (24), and in green (square) the case $L\to x_T$, Eq. (25). The parameters used for the numerical simulations are $D=1$, $r=0.1$, and $x_T=10$. We have used $N={10}^5$ trajectories.

Figure 2

$\frac{D}{x_T^2}{\langle T\rangle }_{x_0}$ as function of the parameter $z=x_T\sqrt{r/D}$ for three values of $L$, $L=1.50$ (red), $L=2.00$ (green), and $L=2.50$ (blue). The lines represent Eq. (27) while the points (circles, triangles, and squares) are data from numerical simulations. The parameters used are $D=1$, $r=0.1$, and $x_T$ is computed given a fixed value of $z$. For the simulations we have used $N={10}^5$ trajectories.

Figure 3

$r_{\text{opt}}{x}_T^2/D$ vs $\epsilon$. The solid line is computed using Eq. (29) while the data points are obtained by numerically computing $d{\langle T\rangle }_{x_0}/dz=0$.

Figure 4

AMFPT for uniformly distributed reset points PDF in an interval $[-L,L]$ for $x_T\in \mathcal{D}$. The solid line represents Eq. (32). The data points are obtained from numerical simulations. The parameters used for the numerical simulations are $D=1$, $r=0.1$, and $x_T=10$. We have used $N={10}^5$ trajectories. In the inset, the solid line represents the AMFPT as a function of the resetting rate $r$ for $L=15$ from Eq. (32). The dashed line shows the predicted scaling as $r^{-1/2}$.

Figure 5

AMFPT for exponentially distributed reset points as a function of $\alpha$. The solid line represents Eq. (35), the data points are obtained from numerical simulations and the dotted line represents the value of ${\langle T\rangle }_{x_0}$ for the EM case [Eq. (24)]. The parameters used for the simulations are $D=1$, $r=0.1$, and $x_T=5$. We have used $N=5\times {10}^5$ trajectories. In the inset, the solid line represents the AMFPT as a function of the resetting rate $r$ for $\alpha =1$ from Eq. (35). The dashed line shows the decaying as $r^{-1/2}$. Both axes are on a logarithmic scale.

Figure 6

Diagram $L\sqrt{r/D}$ vs $p$ for asymmetric resetting points in an interval with $x_T>L$. The parameter $\xi$ is defined as $\xi ={\langle T\rangle }_{x_0}(L=0)/{\langle T\rangle }_{x_0}\left(L\right)$. We have obtained the value of $\xi$ numerically from Eqs. (24) and (37). The green region ($\xi >1$) corresponds to the case where the asymmetric resetting strategy is more efficient, while in the gray region the most efficient strategy is to reset to $x_0=0$. The blue line, computed through Eq. (38), represents $L^{*}\sqrt{r/D}$ vs $p$. Panel (a) is for a reset rate $r=0.01$, panel (b) for $r=0.1$, and panel (c) for $r=1$.

Figure 7

$\frac{D}{x_T^2}{\langle T\rangle }_{x_0}$ as function of the parameter $z=x_T\sqrt{r/D}$ for $p=0.25$ (red) and $p=0.75$ (blue). The lines represent Eq. (41) while the points (circles and triangles) are data from numerical simulations. The parameters used for obtaining the data are $D=1$, $x_T=2$ and $r$ is computed given a fixed value of $z$. In panel (a) $\epsilon =0.2$ thus $L=0.2$. In panel (b) $\epsilon =1.8$ thus $L=3.6$. For the simulations we have used $N={10}^5$ trajectories.

Figure 8

$r_{\mathrm{opt}}\left(\epsilon \right)$ for $\epsilon \to 1$ for $p=0.25$ (red circles) and $p=0.75$ (blue triangles). The dots are obtained by numerically minimizing Eq. (41) and the lines are linear regressions to the data, for each curve the correlation coefficient $R^2=0.9999$. In panel (a) $\epsilon \to 1^{-}$. In panel (b) $\epsilon \to 1^{+}$. The log in the axes stands for the decimal logarithm.

Figure 9

First-passage PDFs. We simulate the resetting process to a point $x_0$ randomly sampled from the symmetric resetting points PDF in the interval $[-L,L]$ and we compute the first-passage PDF, represented in red triangles. We then simulate the resetting process to the ERP and compute the first-passage PDF, represented in blue circles. The $y$ axis is in logarithmic scale, so we can see the exponential behavior of the PDF at long times, as expected from Sec. 2a. The parameters used for the numerical simulations are $x_T=5$, $D=1$, $r=0.1$, $L=2$ and $N={10}^7$ trajectories.

Figure 10

Diagram $x_r/x_T$ vs $L/x_T$ for the equivalent resetting point with $x_T\notin \mathcal{D}$. We have obtained the value of $\mathrm{\Xi }$, defined in Eq. (55), numerically using Eqs. (24) and (23) in panel (a) and Eqs. (24) and (37) in panels (b) and (c). In the gray region ($\mathrm{\Xi }>1$), it is more efficient to reset to distributed random points. However, in the green region, the efficient strategy is to reset to $x_r$. The red solid line represents $x_e^{+}$ while the blue dashed line represents $x_e^{-}$. Panel (a) corresponds to a uniform resetting points PDF in $[-L,L]$ the red and blue lines being computed with Eq. (49). Panels (b) and (c) the searcher resets to $-L$ or $L$ with probabilities $1-p$ and $p$, respectively. Then the red and blue lines are computed with Eq. (50). Panel (b) is for $p=0.25$ and panel (c) is for $p=0.75$.

Figure 11

Diagram $x_r/L$ vs $x_T/L$ for the equivalent resetting point when $x_T\in \mathcal{D}$. We have obtained the value of $\mathrm{\Xi }$, defined in Eq. (55), numerically using Eqs. (24) and (32) in panel (a), Eqs. (24) and (35) in panel (b), and Eqs. (24) and (39) in panels (c) and (d). In the gray region ($\mathrm{\Xi }>1$), it is more efficient to reset to distributed random points. However, in the green region, the efficient strategy is to reset to $x_r$. The red solid line represents $x_e^{+}$ while the blue dashed line represents $x_e^{-}$ computed with Eq. (51). We make use of Eq. (52) in panel (a), Eq. (53) in panel (b), and Eq. (54) in panels (c) and (d).

Figure 12

Coefficient of variation CV as a function of $L$ for $r=0.15$, $r<r_c$, (red solid line and triangles) and $r=0.60$, $r>r_c$ (blue dashed line and circles). The lines represent Eq. (59) while the points are data obtained from numerical simulations. The green horizontal lines represent the EM case. The parameters used are $x_T=8$, $D=1$ and $N={10}^6$ trajectories. $r_c=0.23$.

Figure 13

Coefficient of variation CV as a function of $r$ for $L=5$ (red solid line and triangles) and $L=7$ (blue dashed line and circles). The lines represent Eq. (59) while the points are data obtained from numerical simulations. The parameters used are $x_T=8$, $D=1$, and $N={10}^6$ trajectories. The value of $r$ at which $\mathrm{CV}=1$ is $r_{\text{opt}}=0.044$.

Figure 14

Coefficient of variation CV as a function of $L$ for $r=0.1$ (red solid line and triangles) and $r=0.50$ (blue dashed line and circles). The lines represent Eq. (59) while the points are data obtained from numerical simulations. The parameters used are $x_T=5$, $D=1$ and $N={10}^6$ trajectories.

Figure 15

Coefficient of variation CV as a function of $r$ for $L=10$ (red solid line and triangles) and $L=25$ (blue dashed line and circles). The lines represent Eq. (59) while the points are data obtained from numerical simulations. The parameters used are $x_T=5$, $D=1$, and $N={10}^6$ trajectories.