Fractional Laplacians
A description of my research in nonlocal problems.
In the whole Euclidean space \(\mathbb R^d\).
There are several equivalent representations of fractional Laplacian of order \(s\) in the whole Euclidean space \(\mathbb R^d\):
 Via the Fourier symbol
where \(\mathcal F\) denotes the Fourier transform.
 As a singular kernel operator
 The subdifferential of the \(W^{s,2} (\mathbb R^d) = H^s (\mathbb R^d)\) seminorm given, for \(p>1\), by
An many others (see [1]).
Stochastic interpretation
It is wellknown that the heat equation \(\partial_t \rho = \frac 1 2 \Delta \rho\) can be deduced from Brownian Motion. This motion is constructed by a particle jumping to a grid, to any neighbour node with equal probability. If we replace this simple law by longrange interactions, with a probably of jumping decaying like a power law
\[P(x + hj \to x) = \frac{c(d,s)}{j^{d+2s}}\]Then it is easy to formally recover $u_t + (\Delta)^s u = 0$ (see [2]).
More rigorously, it can be shown that the fractional Laplacian is recovered as the infinitesimal generator of a jump process with powerlaw decay (see, e.g., [3]).
Bounded domains
In a bounded domain \(\Omega\), these different representations give rise to three different operators:
 First, the Spectral Fractional Laplacian (SFL) that has the same eigenfunctions as the usual Laplacian \(\Delta \varphi_i = \lambda_i \varphi_i\) in \(\Omega\) and \(\varphi_i = 0\) on \(\partial \Omega\); and powers of its eigenvalues \(\lambda_i^s\).
This operator can also be constructed from the heat kernel of the usual Laplacian in a bounded domain.

The Restricted Fractional Laplacian (RFL), whose formula is the same as the usual fractional Laplacian. Hence, even though we work on bounded domains the function \(u\) must be defined in the whole space. Thus we impose “exterior conditions” instead of boundary conditions.

Censored Fractional Laplacian (CFL) obtained by integration in the whole domain
The functional treatment of these operators is, in principle, quite different. In particular, the natural spaces of functions must be chosen carefully.
The stochastic interpretation of these three operators can be obtained also from Lévy flights. The difference appears from the treatment of the jumps “across” the boundary of the domain.
References
 Kwasnicki, Mateusz. (2017) Ten equivalent definitions of the fractional laplace operator. Fract. Calc. Appl. Anal.. Link
 Valdinoci, Enrico. (2009) From the long jump random walk to the fractional Laplacian. SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada. Link
 Kühn, Franziska and Schilling, René L.. (2019) On the domain of fractional Laplacians and related generators of Feller processes. J. Funct. Anal.. Link