Lévy 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 $\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.

Resetting-mediated navigation of an active Brownian searcher in a homogeneous topography

“…The targets (S) are assumed to be inside this region and not at the boundaries. Furthermore, the boundary is considered to be reflective, [78][79][80] so that the particles cannot escape and stay confined until the targets are found. The reflecting-boundary condition can be understood in the following way.…”

The Fractional Laplacian with Reflections

“…Reflecting Markov processes may be considered as an instance of stochastic resetting, which is a hot topic in statistical physics concerned, among others, with equilibrium distributions, search optimization, renewal theory and modelling; see, e.g., Evans, Majumdar, and Schehr [30], Garbaczewski and Żaba [34], and Stanislavsky and Weron [63]. Specific reflections, resurrections, or recurrent extensions of stochastic processes with scaling are studied by Kim, Song, and Vondraček [40] for the halfline using the Lamperti transform.…”