Challenging the Lévy Flight Foraging Hypothesis -A Joint Monte Carlo and Numerical PDE Approach
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For a Lévy process on the flat torus 𝕋^2 with power law jump length distribution ∼ |x|^-2-2α for 0<α<1, Monte Carlo and finite difference methods for inverting the fractional Laplacian are employed to confirm recently obtained leading order analytic results for the mean stopping time u_ϵ to a circular target of radius 0 <ϵ≪ 1. The Monte Carlo simulations of the Lévy process rely on a rejection sampling algorithm to sample from a power law distribution, while the finite difference method numerically solves the nonlocal exterior problem (-Δ)^α u_ϵ = -1 with u_ϵ = 0 inside the target. Our results confirm that the mean stopping time indeed scales as O(ϵ^2α-2), in stark contrast to the well-known O(|logϵ|) scaling of the Brownian narrow escape time. For a sufficiently small target size, this difference in scaling implies that a Lévy search strategy may in fact be (significantly) slower than a Brownian search strategy.
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