Extreme statistics of superdiffusive Levy flights and every other Levy subordinate Brownian motion

TL;DR

This paper analyzes the extreme hitting times for superdiffusive Levy flights and Levy subordinate Brownian motions, providing exact distributions and moments, which are crucial for understanding optimal search strategies in complex stochastic systems.

Contribution

It derives the short-time distribution of first hitting times for Levy subordinate Brownian motions and characterizes the extreme FHTs for multiple searchers, extending understanding of superdiffusive search processes.

Findings

Exact distribution of single FHT for Levy subordinate Brownian motion

Distribution and moments of extreme FHTs for many searchers

Numerical validation of theoretical results

Abstract

The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems involve multiple searchers and the important timescale is the time it takes the fastest searcher to find a target, which is called an extreme FHT. In this paper, we study extreme FHTs for any stochastic process that is a random time change of Brownian motion by a Levy subordinator. This class of stochastic processes includes superdiffusive Levy flights in any space dimension, which are processes described by a Fokker-Planck equation with a fractional Laplacian. We find the short-time distribution of a single FHT for any Levy subordinate Brownian motion and use this to find the full distribution and moments of extreme FHTs as the number of searchers grows. We illustrate these rigorous results in several examples and numerical simulations.