Lévy processes in bounded domains: path-wise reflection scenarios and signatures of confinement

Abstract

Abstract We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric α-stable Lévy 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 Lévy process, in terms of the domain of its motion generator, to which an invariant pdf (actually an eigenfunction) does belong.