Lévy flights between absorbing boundaries: Revisiting the survival probability and the shift from the exponential to the Sparre-Andersen limit behavior

We revisit the problem of calculating the survival probability of Lévy flights in a finite interval with absorbing boundaries. Our approach is based on the master equation for discrete Lévy fliers, previously considered to treat the semi-infinite domain. We argue that, although the semi-infinite case can be treated exactly due to Wiener-Hopf factorization, the approximation involved in the problem with the finite interval is actually fairly good. We evidence the shift in the universal behavior of the long-term survival probability from the exponential decay in the presence of two absorbing barriers to the Sparre-Andersen power-law dependence in the single-barrier limit. In some cases, we also calculate the short- and intermediate-term behavior and present the explicit dependence of the survival probability on the Lévy flier's starting position. Our analytical results are confirmed by numerical simulations.

Figure 1

Illustration of a Lévy flight in a finite domain of length $L$, with absorbing boundaries located at $x=-x_0$ and $x=L-x_0$. In this example, after starting from the origin the flier is absorbed by the left boundary as its third jump would lead it to the region $x\le -x_0$.

Figure 2

Survival probability $S\left(t\right)$ as a function of time $t$ for the $\alpha =1$ Cauchy flier in a domain of length (a) $L={10}^2$ and (b) $L={10}^3$, with starting positions $x_0/L=0.002,0.02,0.1$. The long-term exponential behavior, $S\left(t\right)\sim exp(-t/\tau )$, is evidenced both from numerical simulation results (circles), with $x_0$-independent best-fit value ${\tau }^{-1}=2.31/L$, and the analytical approach, with ${\tau }^{-1}=-{\lambda }_{k=1}\approx 3\pi /\left(4L\right)\approx 2.36/L$ [38] (dashed lines).

Figure 3

Survival probability $S\left(t\right)$ as a function of time $t$ for the $\alpha =2$ Gaussian flier. (a) In a domain of length $L={10}^2$, with starting positions $x_0/L=0.01,0.1,0.2$ and $\sigma =1$, the initially faraway boundary located at $x=L-x_0$ can be effectively reached. The long-term exponential behavior, $S\left(t\right)\sim exp(-t/\tau )$, is evidenced from numerical simulation results (circles), with best-fit value ${\tau }^{-1}=9.55/L^2$, as well as from the analytical approach, with ${\tau }^{-1}=-{\lambda }_{k=1}\approx {\pi }^2/L^2\approx 9.87/L^2$ [38] (dashed lines). (b) In a $10\times$ larger domain, $L={10}^3$, with $x_0/L=5\times {10}^{-5},5\times {10}^{-4}$, and $\sigma =1$, the faraway boundary is not effectively reached. The analytical expression for $S\left(t\right)$, Eq. (20), is depicted in dashed lines, and the long-term behavior shifts to the Sparre-Andersen power-law form $S\left(t\right)\sim 1/t^{\nu }$, with $x_0$-independent exponent $\nu =1/2$ (theory) and $\nu =0.49$ (best-fit numerical value).

Figure 4

Integrals $I_{n,\alpha }\left(x\right)$, defined in Eq. (23), as a function of $n$, associated with the Laplace transform of the survival probability, Eq. (22), of a Lévy flier with index $0<\alpha \le 2$. When the argument $x$ is positive and finite, $I_{n,\alpha }\left(x\right)$ presents monotonic decrease with $n$, as seen for $\alpha =0.5,1.0,1.5,2.0$ and $x=1$. Inset: as the argument $x\to \infty$, convergence to the limit $I_{n,\alpha }(x\to \infty )=\pi$ is achieved.