Fractional Laplacians in bounded domains: Killed, reflected, censored and taboo L\'evy flights

TL;DR

This paper explores various definitions and behaviors of fractional Laplacians in bounded domains, analyzing their mathematical properties and physical implications for different boundary conditions and stochastic processes.

Contribution

It provides a comprehensive comparison of boundary-respecting fractional Laplacians, including killed, reflected, censored, and taboo Le9vy flights, highlighting their differences and applications.

Findings

Different boundary conditions lead to inequivalent stochastic processes.

Spectral analysis reveals distinct characteristics for each process type.

Connections to physical systems like disordered semiconductors are discussed.

Abstract

The fractional Laplacian $(- \Delta)^{\alpha /2}$, $\alpha \in (0,2)$ has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of $\alpha $-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 holds true not only for each specific choice of the process behavior at the boundary (like e.g. absorbtion, reflection, conditioning or boundary taboos), but extends as well to its particular technical implementation (Dirchlet, Neumann, etc. problems). The inferred jump-type processes are inequivalent as well, differing in their spectral and statistical characteristics. In the present paper we focus on L\'evy flight-induced jump-type processes which are constrained to stay forever inside a finite domain. That refers to a concept of taboo processes (imported from Brownian to L\'evy - stable contexts), to so-called censored processes and to reflected L\'evy flights whose status still remains to be settled on both physical and mathematical grounds. As a byproduct of our fractional spectral analysis, with reference to Neumann boundary conditions, we discuss disordered semiconducting heterojunctions as the bounded domain problem.