Geodesic L\'evy flights and expected stopping time for random searches

TL;DR

This paper provides an analytic framework for Le9vy flights on various Riemannian manifolds, analyzing their properties and expected search times for small targets, with implications for biological foraging hypotheses.

Contribution

It introduces a detailed analytic description of the infinitesimal generator for Le9vy flights on broad classes of manifolds, advancing understanding of their search efficiency.

Findings

Derived properties of the semigroup associated with Le9vy flights.

Calculated asymptotics of expected stopping times for small targets.

Connected mathematical results to biological foraging hypotheses.

Abstract

We give an analytic description for the infinitesimal generator constructed by Applebaum-Estrade for L\'evy flights on a broad class of closed Riemannian manifolds including all negatively-curved manifolds, the flat torus and the sphere. Various properties of the associated semigroup and the asymptotics of the expected stopping time for L\'evy flight based random searches for small targets, also known as the narrow capture problem, are then obtained using our newfound understanding of the infinitesimal generator. Our study also relates to the L\'evy flight foraging hypothesis in the field of biology as we compute the expected time for finding a small target by using the L\'evy flight random search. A similar calculation for Brownian motion on surfaces was done in [arXiv:2209.12425].