Fractional Laplacians : A short survey

Abstract

This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian. We will first recall some definitions of this operator in R N and its main properties. Then, we will introduce the four main operators often used in the case of a bounded domain ; and we will give several simple and significant examples to highlight the difference between these four operators. Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results. We hope that this short survey will be useful for young researchers of all ages who wish to have a quick idea of the fractional Laplacian(s).

Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement

“…In the present section, we shall outline rudiments of another "reflecting" framework for Lévy processes in the interval, with the fractional heat (Fokker-Planck) dynamics leading asymptotically to the uniform distribution. Its major ingredient is the so-called nonlocal Neumann condition [15][16][17]20]. We note that there are other Neumann condition proposals in existence, [13,14] and [12], but transparent probabilistic pictures, amenable to a computer-assisted (path-wise) verification, appear to be lacking.…”

“…[33,47,48]. Complementary discussions can be found in [20,34,35]. In reference to the interval problems, the pertinent eigenfunctions are bounded and continuous up to the boundaries.…”

“…The original Lévy process is thus replaced by the random walk X(n) of Eq. (20) where Y (i)'s have identical distributions, c.f. Eqs.…”

“…To the contrary, the problem of reflected Lévy processes and their domain-restricted generators, still remains somewhat enigmatic, [1][2][3][4]. Quite apart from the on-going mathematical discussion of (i) appropriate domain restrictions for the fractional Laplacian [12]- [20], (ii) path-wise analysis, mostly based on the Skorohod reflection scenario on the level of stochastic differential equations with the Lévy noise, [22]- [28].…”

“…By making a shift x → x − b/2 (x = 0 is mapped into −b/2, while x = b into +b/2) and next selecting b = 2, we can replace Eq. ( 9) by the form predominantly used in mathematical papers [18,20,33], and likewise in [1,35]. The closed interval of interest [0, b] becomes [−1, 1], and its open version is (−1, 1).…”