Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement

TL;DR

This paper investigates how different path-wise reflection mechanisms affect the confinement and statistical properties of symmetric alpha-stable Levy processes within finite intervals, analyzing their spectral and equilibrium characteristics.

Contribution

It introduces a detailed analysis of various reflection scenarios for Levy processes and explores their impact on the processes' spectral and statistical properties, including invariant densities.

Findings

Reflection mechanisms influence relaxation properties.

Invariant densities depend on reflection type.

Different reflection scenarios alter spectral characteristics.

Abstract

We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric $\alpha $-stable L\'{e}vy processes, whose permanent residence in a finite interval on a line is secured by a two-sided reflection. Depending on the specific reflection "mechanism", the inferred jump-type processes differ in their spectral and statistical characteristics, like e.g. relaxation properties, and functional shapes of invariant (equilibrium, or asymptotic near-equilibrium) probability density functions in the interval. The analysis is carried out in conjunction with attempts to give meaning to the notion of a reflecting L\'{e}vy process, in terms of the domain of its motion generator, to which an invariant pdf (actually an eigenfunction) does belong.