Transition in the decay rates of stationary distributions of L\'evy motion in an energy landscape

TL;DR

This paper investigates a novel transition in the spatial decay behavior of stationary distributions of Lévy processes influenced by external potentials, revealing how process characteristics and eigenvalues affect decay patterns.

Contribution

It uncovers a qualitative transition in decay rates of Lévy process stationary measures caused by changes in process parameters and potential eigenvalues.

Findings

Identification of a transition in decay behavior

Dependence of decay on process and potential parameters

Insights into ground state spatial decay mechanisms

Abstract

The time evolution of random variables with L\'evy statistics has the ability to develop jumps, displaying very different behaviors from continuously fluctuating cases. Such patterns appear in an ever broadening range of examples including random lasers, non-Gaussian kinetics or foraging strategies. The penalizing or reinforcing effect of the environment, however, has been little explored so far. We report a new phenomenon which manifests as a qualitative transition in the spatial decay behavior of the stationary measure of a jump process under an external potential, occurring on a combined change in the characteristics of the process and the lowest eigenvalue resulting from the effect of the potential. This also provides insight into the fundamental question of what is the mechanism of the spatial decay of a ground state.