First Hitting Time of a One-Dimensional Levy Flight to Small Targets

Abstract

First hitting times (FHTs) describe the time it takes a random "searcher" to find a "target" and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the first hitting time to small targets for a one-dimensional superdiffusive search described by a Levy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order $s\in(0,1)$ (describing a $(2s)$-stable Levy flight whose squared displacement scales as $t^{1/s}$ in time $t$) and targets of radius $\varepsilon\ll1$, we show that the MFHT is order one for $s\in(1/2,1)$ and diverges as $\log(1/\varepsilon)$ for $s=1/2$ and $\varepsilon^{2s-1}$ for $s\in(0,1/2)$. We then use our asymptotic results to identify the value of $s\in(0,1]$ which minimizes the average MFHT and find that (a) this optimal value of $s$ vanishes for sparse targets and (b) the value $s=1/2$ (corresponding to an inverse square Levy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations.