Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights.

Abstract

The fractional Laplacian (-Δ)^{α/2}, α∈(0,2), has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of α-stable stochastic processes in R^{n}. On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data-respecting fractional Laplacian should actually be. This ambiguity not only holds true for each specific choice of the process behavior at the boundary (e.g., absorbtion, reflection, conditioning, or boundary taboos), but extends as well to its particular technical implementation (Dirichlet, Neumann, etc., problems). The inferred jump-type processes are inequivalent as well, differing in their spectral and statistical characteristics, which may strongly influence the ability of the formalism (if uncritically adopted) to provide an unambiguous description of real geometrically confined physical systems with disorder. Specifically that refers to their relaxation properties and the near-equilibrium asymptotic behavior. In the present paper we focus on Lévy flight-induced jump-type processes which are constrained to stay forever inside a finite domain. This refers to a concept of taboo processes (imported from Brownian to Lévy-stable contexts), to so-called censored processes, and to reflected Lévy flights whose status still remains to be unequivocally settled. As a by-product of our fractional spectral analysis, with reference to Neumann boundary conditions, we discuss disordered semiconducting heterojunctions as the bounded domain problem.