Random searches with finite-lived targets

Random searches find applications in a broad range of problems in biology, chemistry or computer sciences.  In a typical set-up, a searcher follows a random motion and reacts with a fixed target site upon first encounter. In many situations, however, the target has a finite life-time after which it becomes inactive, lost or no longer available for reaction.  To be successful, the searcher must hence find the target before the latter becomes inactive. We present exact results in one dimension when the searcher performs a discrete time random walk or a Lévy flight. In contrast with the case of a permanent target, if one wishes to optimize the capture probability or minimize the conditional mean first passage time, the Lévy index must take a non-trivial value, which exhibits a "phase transition" as the initial distance to the target site is varied.  We outline connections between this problem and search processes based on resetting.  

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