Dynamics of confined Levy flights in terms of (Levy) semigroups

TL;DR

This paper explores the mathematical structure of confined Levy flights using semigroup theory, addressing the reconstruction of invariant probability densities for all stability indices and discussing related challenges.

Contribution

It introduces a method for reconstructing ground states of Levy semigroups for all stability indices, enhancing understanding of confined Levy flight dynamics.

Findings

Reconstruction of invariant PDFs for Levy flights across all stability indices.

Identification of conditions when invariant PDFs are truly asymptotic.

Analysis of the behavior of PDFs at the boundaries of the stability interval.

Abstract

The master equation for a probability density function (pdf) driven by L\'{e}vy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic L\'{e}vy-stable semigroups, for {\em all} values of the stability index $\mu \in (0,2)$. That is known to resolve an invariant pdf for confined L\'{e}vy flights (e.g. the former jump-type process). Jeopardies of the procedure are discussed, with a focus on: (i) when an invariant pdf actually is an asymptotic one, (ii) subtleties of the pdf $\mu $-dependence in the vicinity and sharply {\em at} the boundaries 0 and 2 of the stability interval, where jump-type scenarios cease to be valid.