First passage under stochastic resetting in an interval

We consider a Brownian particle diffusing in a one-dimensional interval with absorbing end points. We study the ramifications when such motion is interrupted and restarted from the same initial configuration. We provide a comprehensive study of the first-passage properties of this trapping phenomena. We compute the mean first-passage time and derive the criterion on which restart always expedites the underlying completion. We show how this set-up is a manifestation of a success-failure problem. We obtain the success and failure rates and relate them with the splitting probabilities, namely the probability that the particle will eventually be trapped on either of the boundaries without hitting the other one. Numerical studies are presented to support our analytic results.

Figure 1

Schematic of a Brownian particle diffusing in a bounded domain $[a,b]$ in the presence of resetting which reinstates the particle at its initial position $x_0$ with a rate $r$.

Figure 2

Plot for the unconditional mean first-passage time as a function of restart rate for two different boundary conditions: (i) $a=0,b=3$ and (ii) $a=0,b=5$, with the initial condition fixed at $x_0=1$ for both cases. The diffusion constant has the value $D=1/2$. The theoretical formulas [dashed and solid lines for (i) and (ii) respectively], as in Eq. (15), are corroborated by their respective simulation data (in markers).

Figure 3

Plot for the scaling function $\mathcal{G}(\beta ,u)$ [Eq. (19)] as a function of $\beta$. The function has minima when $u$ takes value from the domain $\mathcal{D}$, as shown for $u=0.2<u_{-}$ (short dashed line in the middle) and $u=0.85>u_{+}$ (long dashed line at the bottom). However, when $u_{-}<u=0.65<u_{+}$ (solid line at the top), there is no minimum for $\mathcal{G}(\beta ,u)$ at any finite $r$, indicating that restart is not beneficial.

Figure 4

Depiction of first passage under resetting in an interval as a success-failure problem. A “successful” event is defined by an absorption at the boundary “$b$,” while a “failure” event occurs when an absorption takes place at the boundary “$a$.”

Figure 5

We have plotted the conditional exit times ${\langle t\rangle }_{+}$ and ${\langle t\rangle }_{-}$ as a function of restart rate $r$ for a system with the end points $a=0,b=3$ and resetting position $x_0=1$ with $D=1/2$. The dashed and solid lines represent analytical expressions for ${\langle t\rangle }_{+}$ and ${\langle t\rangle }_{-}$ [given by Eq. (26)], respectively. Markers in circle and triangle represent the corresponding simulation data points. The inset shows the plot for the function ${\varepsilon }_{+}{\langle t\rangle }_{+}+{\varepsilon }_{-}{\langle t\rangle }_{-}$ (dashed line) against $\langle T_r\rangle$ (markers) as a function of restart rate. The match between the two provides a numerical demonstration of Eq. (33).

Figure 6

Plot of splitting probabilities ${\varepsilon }_{+}\left(x_0\right)$ and ${\varepsilon }_{-}\left(x_0\right)$ as a function of restart rate (in solid and dashed lines, respectively). The parameters are taken as $a=0,b=3,x_0=2$, and $D=1/2$.

Figure 7

Plot for the conditional mean exit times ${\langle t\rangle }_{\pm }$ (in long and short dashed lines, respectively) and the unconditional mean exit time $\langle T_r\rangle$ (in solid line) as a function of the resetting position $x_0$. The boundaries are taken at $a=0,b=3$ for the parameter values: $D=0.5, r=0.3$. The times coincide for the symmetric case (when $x_0=3/2$), as pointed out in the text.

Figure 8

Plot of the slope $\mathcal{S}(x_0,r)$ as a function of $x_0$ (where $a=0,b=3$) for three different values of restart rate $r=0.01$ (short dashed line), $r=0.5$ (long dashed line), and $r=3.0$ (solid line). The slope $\mathcal{S}(x_0,r)$ becomes zero at $x_0=a$ when $r=0.01$. However, when $r=0.5$, the slope can be both positive or negative with a zero at $x_0$ ($a<x_0<\frac{a+b}{2}$). The behavior remains identical when $r=3.0$ except that the function $\mathcal{S}(x_0,r)$ has a zero at $x_0=\frac{a+b}{2}$, which is the middle of the interval. In the inset we have plotted ${\langle t\left(x_0\right)\rangle }_{+}$ as a function of $x_0$ for the same set of values of $r$ mentioned above. The plot conforms the same behavior as mentioned above.