Rare events in generalized L\xe9vy Walks and the Big Jump principle

Alessandro Vezzani,Raffaella Burioni,Eli Barkai

doi:10.1038/s41598-020-59187-w

Alessandro Vezzani, Raffaella Burioni + Show 1 more

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https://doi.org/10.1038/s41598-020-59187-w

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Abstract

The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic processes with power law distributed step durations and with complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.

Highlights

Rare events are an important and exciting theoretical research field in mathematics and in natural sciences, with a long history in topics ranging from physics, geophysics and biology, to ecology and social systems[1,2,3,4]
The duration of a step t is drawn from a power law distribution λ(t) ∼ t−1−α, while the motion within a step is described by two further exponents: v, relating the step length with the duration time t, and η, which provides the temporal dynamics within a step, modelling acceleration and deceleration effects
The estimate is based on the splitting of the problem in two parts: the first one leads to the calculation of the jump rate, that is the rate at which the walker makes attempts to perform the big jump

Summary

Introduction

Rare events are an important and exciting theoretical research field in mathematics and in natural sciences, with a long history in topics ranging from physics, geophysics and biology, to ecology and social systems[1,2,3,4]. The big jump principle was recently extended[15,17,18] to case studies which involve Lévy walks These are introduced as continuous time stochastic process for particles performing steps with duration drawn from a power law, heavy-tailed, distribution[19,20,21]. Because of their generality, Lévy walks are applied to describe motion of cold atoms in laser cooling[22], transport in turbulent flow[23] and in neural transmission[24], animal motion[25,26], and natural and optimized search processes[27]. We describe the asymptotic time evolution of the entire walker probability distribution, which allows us to extract the behavior of correlations and higher moments

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