Rare Events in Extreme Value Statistics of Jump Processes with Power Tails

TL;DR

This paper analyzes the probability of rare, extreme events in jump processes with power-law tails, revealing how large fluctuations are typically caused by single large jumps and exploring their implications across various scientific fields.

Contribution

It provides analytical forms for the tail distributions of maxima in Levy flights, Levy walks, and Levy-Lorentz gas, extending previous results and highlighting the role of lattice topology in extreme value statistics.

Findings

Re-derivation of recent results for Levy flights using big-jump principle

Identification of memory effects in Levy-Lorentz gas due to lattice topology

Confirmation of analytical results through extensive numerical simulations

Abstract

We study rare events in the extreme value statistics of stochastic symmetric jump processes with power tails in the distributions of the jumps, using the big-jump principle. The principle states that in the presence of stochastic processes with power tails statistics, if at a certain time a physical quantity takes on a value much larger than its typical value, this large fluctuation is realised through a single macroscopic jump that exceeds the typical scale of the process by several orders of magnitude. In particular, our estimation focuses on the asymptotic behaviour of the tail of the probability distribution of maxima, a fundamental quantity in a wide class of stochastic models used in chemistry to estimate reaction thresholds, in climatology for earthquake risk assessment, in finance for portfolio management, and in ecology for the collective behaviour of species. We determine the analytical form of the probability distribution of rare events in the extreme value statistics of three jump processes with power tails; L\'evy flights, L\'evy walks and the L\'evy-Lorentz gas. For the L\'evy flights, we re-obtain through the big-jump approach recent analytical results, extending their validity. For the L\'evy-Lorentz gas we show that the topology of the disordered lattice along which the walker moves induces memory effects in its dynamics, which influences the extreme value statistics. Our results are confirmed by extensive numerical simulations.