Lévy flights and Lévy walks under stochastic resetting

Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. We demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy-type, $\alpha$-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index $\alpha$ determining the power-law decay of the jump length distribution. For heavier (smaller $\alpha$) distributions, the domain becomes narrower in comparison to lighter tails. Additionally, we explore connections between Lévy flights (LFs) and Lévy walks (LWs) in the presence of stochastic resetting. First of all, we show that for Lévy walks, the stochastic resetting can also be beneficial in the domain where the coefficient of variation is smaller than 1. Moreover, we demonstrate that in the domain where LWs are characterized by a finite mean jump duration (length), with the increasing width of the interval, the LWs start to share similarities with LFs under stochastic resetting.

Figure 1

Numerically estimated MFPT (dots) and CV (squares) as a function of the initial condition $x_0$ for $\alpha =2$ along with theoretical values (solid lines). The gray region shows the domain where $\mathrm{CV}\left(x_0\right)>1$.

Figure 2

MFPT under stochastic (Poissonian) restarting for the GWN ($\alpha =2$) driving as a function of the resetting rate $r$. Various lines correspond to different initial conditions $x_0$. Points represent results of numerical simulations, while solid lines represent results of theoretical predictions given by Eq. (10).

Figure 3

(a) Numerically estimated $\mathrm{CV}\left(x_0\right)$ for $\alpha \in \{0.5,1.0,1.5,2.0\}$ with $L=1$ (points) complemented by theoretical results (solid lines); see Eqs. (1, 2, 3). The dashed line shows the $\mathrm{CV}=1$ level. (b) Numerically calculated $x_{\mathrm{tr}}\left(\alpha \right)$ such that $\mathrm{CV}\left(x_{\mathrm{tr}}\right)=1$ (solid line) and the width $W\left(x_0\right)$ of the domain where resetting is beneficial (dot-dashed line); see Eq. (11). For $x_0>x_{\mathrm{tr}}$, the coefficient of variation is greater than 1. The dashed line indicates the GWN limit, i.e., $x_{\mathrm{tr}}=1/\sqrt{5}$.

Figure 4

MFPT under Poissonian resetting. Various points represent different values of the stability index $\alpha$. The solid line depicts theoretical dependence for $\alpha =2$ [see Eq. (10)], while dashed lines depict $\mathcal{T}(x_0,r=0)$ [see Eq. (2)]. The initial condition is set to (a) $x_0=0.5$ and (b) $x_0=0.7$.

Figure 5

Coefficient of variation, CV [see Eq. (1)], as a function of the initial position $x_0$. Points represent the CV for LWs (obtained through numerical simulations) and lines represent the CV for LFs [analytical results from Eqs. (2) and (3)]. Various panels correspond to different values of the interval half-width $L$: (a) $L=1$ and (b) $L=1000$.

Figure 6

Numerically estimated (analogous to Fig. 5) coefficient of variation, CV (lines) [see Eq. (1)], as a function of the initial position $x_0$ for LWs along with the normalized ratio $\mathrm{\Lambda }\left(x_0\right)$ (points); see Eq. (14). Various panels correspond to different values of the interval half-width $L$: (a) $L=1$, (b) $L=100$, and $L=10000$ [inset in (b)].