Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
Abstract
We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric -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.
Keywords
Cite
@article{arxiv.2201.09582,
title = {Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement},
author = {Piotr Garbaczewski and Mariusz Żaba},
journal= {arXiv preprint arXiv:2201.09582},
year = {2022}
}
Comments
20 pp, 8 figures, Text amendments, Abstract and Section I modified