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First hitting times (FHTs) describe the time it takes a random "searcher" tofind a "target" and are used to study timescales in many applications. FHTshave been well-studied for diffusive search, especially for small targets,which is called the narrow capture or narrow escape problem. In this paper, westudy the first hitting time to small targets for a one-dimensionalsuperdiffusive search described by a Levy flight. By applying the method ofmatched asymptotic expansions to a fractional differential equation we obtainan explicit asymptotic expansion for the mean FHT (MFHT). For fractional order$s\in(0,1)$ (describing a $(2s)$-stable Levy flight whose squared displacementscales as $t^{1/s}$ in time $t$) and targets of radius $\varepsilon\ll1$, weshow 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 $s\in(0,1]$ whichminimizes the average MFHT and find that (a) this optimal value of $s$ vanishesfor sparse targets and (b) the value $s=1/2$ (corresponding to an inversesquare Levy search) is optimal in only very specific circumstances. We confirmour results by comparison to both deterministic numerical solutions of theassociated fractional differential equation and stochastic simulations.

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