Searching with and against the stream: Levy or Brown?

TL;DR

This paper compares Levy flights and Brownian motion in search tasks with external drift, revealing conditions where each is optimal and challenging the belief that alpha=1 Levy flights are always best.

Contribution

It establishes criteria for when Levy flights outperform Brownian motion in drifted environments and shows the optimal Levy index can vary across (1,2), including Brownian motion.

Findings

Levy flights are efficient when the target is upstream.

Brownian motion is better when the target is downstream.

Optimal Levy index can be in (1,2), not just 1.

Abstract

We study the efficiency of search processes based on Levy flights (LFs) with power-law distributed jump lengths in the presence of an external drift. While LFs turn out to be efficient search processes when relative to the starting point the target is upstream, in the downstream scenario regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of LFs, due to which LFs typically overshoot a point in space. We establish criteria when the combination of the external stream and the initial distance between the starting point and the target favors LFs over regular Brownian search. Contrary to the common belief that LFs with a stable index alpha=1 are optimal, we find that the optimal alpha may range in the entire interval (1,2) and even include Brownian motion as the overall most efficient search strategy.