Levy flights and Levy -Schroedinger semigroups

TL;DR

This paper investigates two mechanisms confining Levy flights under external potentials, demonstrating the existence of topological Levy processes sharing stationary distributions with Langevin-based models, thus advancing understanding of Levy flight dynamics.

Contribution

It introduces and compares two confining mechanisms for Levy flights, highlighting the role of Levy-Schroedinger semigroups in generating topological Levy processes with matching invariant distributions.

Findings

Existence of topological Levy processes with the same invariant pdf as Langevin-based models.

Demonstration of reverse correspondence between the two confining mechanisms.

Insight into the relationship between different Levy flight confinement methods.

Abstract

We analyze two different confining mechanisms for L\'{e}vy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological L\'{e}vy process with the very same invariant pdf and in the reverse.