Fractional Laplacians

A description of my research in non-local 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
\[(-\Delta)^s u (x) = \mathcal F^{-1} \left[ |\xi|^{2s} \mathcal F[u] \right] (x)\]

where \(\mathcal F\) denotes the Fourier transform.

  • As a singular kernel operator
\[(-\Delta)^s u (x) = c_{n,s} \lim_{\varepsilon \to 0} \int_{\mathbb R^d \setminus B_\varepsilon (x) } \frac{u(x) - u(y)}{|x-y|^{n+2s}} {\mathrm d} y,\]
  • The subdifferential of the \(W^{s,2} (\mathbb R^d) = H^s (\mathbb R^d)\) semi-norm given, for \(p>1\), by
\[[u]_{W^{s,p}} = \left( \int_{\mathbb R^d } \int_{\mathbb R^d} \frac{|u(x) - u(y)|^p}{|x-y|^{n+sp}} {\mathrm d} x\,{\mathrm d} y\right)^{\frac 1 p}.\]

An many others (see [1]).

Stochastic interpretation

It is well-known 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 long-range 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 power-law 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 \(\lambda_i^s\).
\[(-\Delta)^s_{SFL} u (x) = \sum_{i=1}^\infty \lambda_i^{s} \langle u, \varphi_i \rangle .\]

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

\[(-\Delta)^s_{CFL} u = c_{n,s} \lim_{\varepsilon \to 0} \int_{\Omega \setminus B_\varepsilon (x) } \frac{u(x) - u(y)}{|x-y|^{n+2s}} {\mathrm d} y.\]

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

  1. Kwasnicki, Mateusz. (2017) Ten equivalent definitions of the fractional laplace operator. Fract. Calc. Appl. Anal.. Link
  2. Valdinoci, Enrico. (2009) From the long jump random walk to the fractional Laplacian. SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada. Link
  3. Kühn, Franziska and Schilling, René L.. (2019) On the domain of fractional Laplacians and related generators of Feller processes. J. Funct. Anal.. Link