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We study the efficiency of random search processes based on L{\'e}vy flightswith power-law distributed jump lengths in the presence of an external drift,for instance, an underwater current, an airflow, or simply the bias of thesearcher based on prior experience. While L\'evy flights turn out to beefficient search processes when relative to the starting point the target isupstream, in the downstream scenario regular Brownian motion turns out to beadvantageous. This is caused by the occurrence of leapovers of L{\'e}vyflights, due to which L{\'e}vy flights typically overshoot a point or smallinterval. Extending our recent work on biased LF search [V. V. Palyulin, A. V.Chechkin, and R. Metzler, Proc. Natl. Acad. Sci. USA,DOI:10.1073/pnas.1320424111] we establish criteria when the combination of theexternal stream and the initial distance between the starting point and thetarget favors L{\'e}vy flights over regular Brownian search. Contrary to thecommon belief that L{\'e}vy flights with a L{\'e}vy index $\alpha =1$ (i.e.,Cauchy flights) are optimal for sparse targets, we find that the optimal valuefor $\alpha$ may range in the entire interval $(1,2)$ and include Brownianmotion as the overall most efficient search strategy.

Optimization of random search processes in the presence of an external bias