Fractional Laplacians and Levy flights in bounded domains

Abstract

We address Lévy-stable stochastic processes in bounded domains, with a focus on a discrimination between inequivalent proposals for what a boundary data-respecting fractional Laplacian (and thence the induced random process) should actually be. Versions considered are: restricted Dirichlet, spectral Dirichlet and regional (censored) fractional Laplacians. The affiliated random processes comprise: killed, reflected and conditioned Lévy flights, in particular those with an infinite life-time. The related concept of quasi-stationary distributions is briefly mentioned. I. MOTIVATIONJump-type Lévy processes in a bounded domain are a subject of an active study both in physics and mathematics communities. The physics-oriented research is conducted with some disregard to an ample coverage of the topic in the past and modern mathematical literature. The reason is rooted not only in a methodological gap between the practitioners' pragmatism and the mathematically rigorous reasoning. An important factor is a scarce (or even lack of) communication between various research groups and research streamlines. This refers not only to rather residual physics-mathematics interplay, but also to the mathematics community per se: relevant publications are scattered in a large number of highly specialized journals and easily escape the attention of potentially interested parties.Recently an attempt has been made to establish a common conceptual basis for varied frameworks in which fractional Laplacians appear. Formally looking different, but actually equivalent, definitions of fractional Laplacians, appropriate for the description of Lévy stable processes in R n , n ≥ 1, have been collected and their mutual relationships analyzed in minute detail in Ref. [1].There is a general consensus that the standard Fourier multiplier definition appears to be defective, if one passes to Lévy flights in a bounded domain. This is a consequence of an inherent nonlocality of Lévy-stable generators. Different proposals for boundary-data-respecting fractional Laplacians were given in the literature. Often, with a view towards more efficient computer-assisted calculations (that mostly in connection with nonlinear fractional differential equations, [11,12]).However, in contrast to the situation in R n , these proposals are known to be inequivalent, c.f. Refs.[2]- [16], see also [18,19]. Likewise, the induced jump-type processes are inequivalent and have different statistical characteristics. That in particular refers to a standard physical inventory, adapted directly from the Brownian motion studies [20]: the statistics of exits from the domain, e.g. first and mean first exit times, probability of survival, its long time behavior an ultimate asymptotic decay [21]- [27].Interestingly, the existence problem for jump-type processes with an infinite life-time in a bounded domain, seems to have been left aside in the physics literature (compare e.g. Ref. [28] in connection with diffusion processes and Ref. [30] for a preliminary discussion of the Cauchy process...

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