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

Abstract

The fractional Laplacian (−∆) 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 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, which may strongly influence the ability of the formalism (if uncritically adopted) to provide an unambigous 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. That 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 byproduct of our fractional spectral analysis, with reference to Neumann boundary conditions, we discuss disordered semiconducting heterojunctions as the bounded domain problem. I. MOTIVATIONBrownian motion in a bounded domain is a classic problem with an ample coverage in the literature, specifically concerning the absorbing (Dirichlet) and reflecting (Neumann) boundary data (for the present purpose we disregard other boundary data choices). A coverage concerning their physical relevance is enormous as well [1,2].Anticipating further discussion, we quite inentionally point out source papers dealing with reflected Brownian motion, [3] and exposing at some length the method of eigenfuction expansions for the reflected and other boundarydata problems, [4]- [8], c.f. also [9]. The latter method is as well an indispensable tool in the analysis of spectral properties of fractional Laplacians and related jump-type processes in bounded domains. Its direct link with well developed theory of heat semigroups for jump-type processes (mostly these with absorpion/killing) allows to address the statistics of exits from the domain, like e.g. the first and mean first exit times, large time behavior, stationarity issues, probability of survival and its asymptotic decay, c.f.[10]- [15]. Compare e.g. also [16,17] (Brownian case) and [19][20][21] (Lévy-stable case), where the role of lowest eigenstates and eigenvalues (thence eigenvalue gaps) of the motion generator has appeared to be vital for the description of decay rates of killed stochastic processes. The spectral data of motion generators are relevant for quantifying long-living processes in a spatial...

Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

“…In fact, such two kinds of definitions are not equivalent in high-dimensional cases. 11,15,16 It means that the "direct" study of time-space fractional diffusion models with the IFL should be worthily considered.…”

“…From a probabilistic point of view, the IFL represents the infinitesimal generator of a symmetric -stable Lévy process. 11,16,18 Mathematically, the well-posedness/regularity of the Cauchy problem or uniqueness of the solutions of the TSFDE (1.1) has been studied in previous works. 3,[19][20][21][22][23] Due to the nonlocality, the analytical (or closed-form) solutions of TSFDEs (1.1) on a finite domain are rarely available.…”

“…14 However, such a numerical framework cannot be directly extended to solve the two-and three-dimensional TSFDEs due to the IFL (−Δ) ∕2 u(x, ) ≠ − u (x, ) |x| − u (x, ) | | . 11,15,16 Therefore, it will hinder the development of numerical solutions for TSFDEs from the stated objective.…”

“…On the other hand, although most of time–space fractional diffusion models are initially defined with the spatially integral fractional Laplacian (IFL), 9‐12 many previous studies (cf., e.g., previous works 4,5,13‐15 ) always substitute the space Riesz fractional derivative 1 for the IFL. In fact, such two kinds of definitions are not equivalent in high‐dimensional cases 11,15,16 . It means that the “direct” study of time–space fractional diffusion models with the IFL should be worthily considered.…”

“…The normalization constant c 1, α is defined as with Γ(·) denoting the gamma function. From a probabilistic point of view, the IFL represents the infinitesimal generator of a symmetric α ‐stable Lévy process 11,16,18 . Mathematically, the well‐posedness/regularity of the Cauchy problem or uniqueness of the solutions of the TSFDE () has been studied in previous works 3,19‐23 …”