First Hitting Time of a One-Dimensional Lévy Flight to Small Targets
Daniel Gomez, Sean D. Lawley
Source abstract
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 FHT to small targets for a one-dimensional superdiffusive search described by a Lévy 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 [Formula: see text] (describing a [Formula: see text]-stable Lévy flight whose squared displacement scales as [Formula: see text] in time [Formula: see text]) and targets of radius [Formula: see text], we show that the MFHT is order one for [Formula: see text] and diverges as [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text]. We then use our asymptotic results to identify the value of [Formula: see text] which minimizes the average MFHT and find that (a) this optimal value of [Formula: see text] vanishes for sparse targets and (b) the value [Formula: see text] (corresponding to an inverse square Lévy 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.
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