L\'{e}vy flights in inhomogeneous environments

TL;DR

This paper investigates the long-term behavior of Lévy flights in various potentials using semigroup models, revealing stronger confinement effects and establishing criteria for invariant probability densities in jump processes.

Contribution

It introduces a semigroup-based approach to analyze Lévy flights in inhomogeneous environments, providing new criteria for invariant densities and comparing with Langevin models.

Findings

Semigroup models exhibit stronger confining properties than Langevin models.

Established verifiable criteria for invariant probability densities.

Illustrated results with Cauchy driver and polynomial potentials.

Abstract

We study the long time asymptotics of probability density functions (pdfs) of L\'{e}vy flights in different confining potentials. For that we use two models: Langevin - driven and (L\'{e}vy - Schr\"odinger) semigroup - driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. Since contractive semigroups set a link between L\'{e}vy flights and fractional (pseudo-differential) Hamiltonian systems, we can use the latter to control the long - time asymptotics of the pertinent pdfs. To do so, we need to impose suitable restrictions upon the Hamiltonian and its potential. That provides verifiable criteria for an invariant pdf to be actually an asymptotic pdf of the semigroup-driven jump-type process. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to L\'{e}vy oscillators), with respect to both dynamical mechanisms.