Fractional Laplacian Overview

• Fractional Laplacian is a nonlocal pseudo-differential operator defined via equivalent constructions such as Fourier multipliers, singular integrals, and spectral methods.
• It underpins models for anomalous diffusion, stable Lévy processes, and nonlocal PDEs, playing a crucial role in various scientific and applied fields.
• Numerical methods including finite difference schemes, Galerkin FEM, and deep neural networks are employed to approximate the operator effectively.

The fractional Laplacian is a canonical example of a nonlocal pseudo-differential operator that generalizes the classical Laplacian $-\Delta$ to fractional (and, more generally, non-integer) exponents. It arises naturally in the theory of stable Lévy processes, anomalous diffusion, nonlocal PDEs, and the analysis of function spaces. Multiple equivalent formulations exist in the whole space, while in bounded domains the choice of definition affects both analytical and numerical properties, reflecting the fundamentally nonlocal character of the operator.

1. Definitions and Equivalent Constructions

Several equivalent definitions exist for the fractional Laplacian $(-\Delta)^s$ on $\mathbb{R}^n$, $0

• Fourier Multiplier Definition: For $u\in\mathcal{S}(\mathbb{R}^n)$,

$$(-\Delta)^s u(x) = \mathcal{F}^{-1} \bigl[ |\xi|^{2s} \hat{u}(\xi) \bigr](x),$$

where $\mathcal{F}$ denotes the Fourier transform. The operator is thus a pseudo-differential operator with symbol $|\xi|^{2s}$ (Teymurazyan, 2023, Pagnini et al., 2023, Lischke et al., 2018).

• Singular-Integral (Riesz) Definition:

$$(-\Delta)^s u(x) = C_{n,s}~ \text{P.V.} \int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,$$

with normalization $$C_{n,s} = \frac{2^{2s}s\,\Gamma(\frac{n+2s}{2})}{\pi^{n/2}\Gamma(1-s)}$$ (Teymurazyan, 2023, Lischke et al., 2018, Michelitsch et al., 2015, Dyda, 2011). The singularity at $x=y$ is handled in the Cauchy principal value sense.

• Semigroup (Bochner/Heat Kernel) Definition:

$$(-\Delta)^s u(x) = \frac{1}{\Gamma(-s)} \int_0^\infty \big(e^{t\Delta}u(x) - u(x)\big)\, t^{-1-s}\,dt,$$

relating the fractional Laplacian to a subordinated Brownian motion (Teymurazyan, 2023, Pagnini et al., 2023).

• Spectral Definition (on bounded domains): If $u$ admits an expansion $u = \sum_k u_k \phi_k$ in terms of Dirichlet eigenfunctions $\phi_k$ with $-\Delta \phi_k = \lambda_k \phi_k$, then

$$(-\Delta)^s_{\Omega} u = \sum_k \lambda_k^s u_k \phi_k,$$

which is the spectral fractional Laplacian (Teymurazyan, 2023, Harizanov et al., 2020, Lischke et al., 2018).

• Mellin Transform Definition (for radially symmetric $u$): For $u(x)=f(|x|)$,

$$\mathcal{M}\{(-\Delta)^{\alpha/2} f\}(s) = -2^{\alpha} \frac{\Gamma(s)\Gamma\left(\frac{n - (s-\alpha)}{2}\right)}{\Gamma\left(\frac{n-s}{2}\right)} \mathcal{M}\{f\}(s-\alpha),$$

with inverse Mellin representation (Pagnini et al., 2023).

All these are equivalent on $\mathcal{S}(\mathbb{R}^n)$ or sufficiently decaying and regular functions (Teymurazyan, 2023, Pagnini et al., 2023).

2. Analytical Properties and Functional Framework

Key analytical properties follow directly from the above definitions:

• Linearity, Self-Adjointness, Positivity: $(-\Delta)^s$ is linear and self-adjoint on $L^2(\mathbb{R}^n)$, with spectrum $[0,\infty)$ (Teymurazyan, 2023, Lischke et al., 2018).
• Scaling: For $u_\lambda(x) = u(\lambda x)$,

$$(-\Delta)^s u_\lambda(x) = \lambda^{2s} \left[(-\Delta)^s u\right](\lambda x)$$

(Zheng et al., 2023, Teymurazyan, 2023).

• Nonlocality: $(-\Delta)^s u(x)$ depends on values of $u$ at points $y$ arbitrarily far from $x$; it encodes effects of long-range interactions not present in the classical $\Delta$ (Lischke et al., 2018, Teymurazyan, 2023).
• Maximum Principle: For $s\in(0,1)$ and suitable exterior sign conditions, $(-\Delta)^s$ satisfies a nonlocal strong maximum principle (Teymurazyan, 2023, Lischke et al., 2018).
• Regularity Theory: Solutions to $(-\Delta)^s u = f$ inherit regularity depending on $f$ and $s$. For $f\in C^\alpha$ one gains $u \in C^{2s+\alpha}$ up to the loss due to boundary layers and nonlocality (Teymurazyan, 2023, Harizanov et al., 2020, Lischke et al., 2018).

3. Boundary Value Problems and Distinction Between Notions

On bounded domains $\Omega$, at least two fundamentally different notions of the fractional Laplacian emerge; this reflects different nonlocal boundary constraints (Lischke et al., 2018, Harizanov et al., 2020):

• Riesz (Integral) Fractional Laplacian:

$$(-\Delta)^s u(x) = C_{n,s}~ \text{P.V.} \left[\int_{\Omega} \frac{u(x)-u(y)}{|x-y|^{n+2s}} dy + \int_{\mathbb{R}^n\setminus\Omega} \frac{u(x)-g(y)}{|x-y|^{n+2s}} dy \right], \quad x\in\Omega$$

with "volume constraint" $u=g$ in $\mathbb{R}^n\setminus\Omega$. This is the generator of the killed $\alpha$-stable process (Lischke et al., 2018, D'Elia et al., 2013).

• Spectral Fractional Laplacian:

Defined via the eigenfunction expansion with homogeneous Dirichlet (or Neumann) boundary conditions on $\partial \Omega$ alone. For inhomogeneous boundary data, a harmonic lifting (i.e., $u = v + w$ with $v$ harmonic and matching $g$ at the boundary, $w|_{\partial\Omega} = 0$) can be used (Harizanov et al., 2020, Lischke et al., 2018).

These two definitions are not equivalent: solutions exhibit distinct interior and boundary layer behaviors, and obey different stochastic interpretations (killed jump process vs. subordinate stopped Brownian motion) (Lischke et al., 2018, Harizanov et al., 2020).

4. Generalizations: Variable-Order, Anisotropic, and Generalized Operators

• Variable-Order Fractional Laplacian (VOFL): For $s : \mathbb{R}^n \rightarrow (0, n/2)$, especially radial $s(x)$, the VOFL is constructed as the inverse of a space-dependent Riesz potential,

$(-\Delta)^{s(\cdot)} = I_{2s(\cdot)}^{-1}$

with $$I_{2s(\cdot)} f(x) = \int_{\mathbb{R}^n} K_{s(\cdot)}(x-y) f(y)\, dy$$ and

$$K_{s(\cdot)}(x) = \Gamma\left(\frac{n}{2} - s(|x|)\right) / \big[ 4^{s(|x|)} \pi^{n/2} \Gamma(s(|x|)) \big]\, |x|^{-n+2s(|x|)}$$

(Darve et al., 2021). Analytical properties like linearity, invertibility, and rotation-invariance remain, but scaling holds only locally.

• Spatially Variant Fractional Laplacian: For $s(\cdot)$ measurable (possibly in $[0,1]$), the operator can be characterized variationally as a Dirichlet-to-Neumann map of a variable-weight degenerate extension over $\Omega\times(0,\infty)$. This allows the well-posed definition and trace regularity in weighted Sobolev/Besov spaces (Ceretani et al., 2021).
• Generalized Fractional Laplacian in Nonhomogeneous Medium: In $\mathbb{R}^2$, write

$(-\Delta)^{s} u = \nabla \cdot I_{2-2s} \nabla u$

where $I_{2-2s}$ is the Riesz potential operator. Generalization to a variable positive-definite matrix field $K(x)$ yields

$$L_{s,K} u = \nabla \cdot \left[ K(x) I_{2-2s} (K(x) \nabla u) \right]$$

suitable for modeling anomalous diffusion in nonhomogeneous media (Zheng et al., 2023).

5. Numerical Methods: Discretization and Computational Algorithms

Several classes of numerical methods are effective for approximating the fractional Laplacian:

Scheme	Domain	Main Feature	Typical Accuracy
Finite difference/gl	$\mathbb{R}$, grids	Discrete convolution, explicit weights	$O(h^{2-\alpha})$ to $O(h^{3-\alpha})$ (Huang et al., 2013, Huang et al., 2016)
Truncated quadrature	$\mathbb{R}$, $\mathbb{R}^d$	Directly approximates PV integral	2$^{\text{nd}}$ order (Cayama et al., 2022, Minden et al., 2018)
FFT-based convolution	$\mathbb{R}^d$ / periodic	Exploits translation invariance	Fast (FFT), 2$^{\text{nd}}$ order (Minden et al., 2018, Cayama et al., 2022)
Galerkin FEM/adaptive AFEM	Bounded domains	Variational, dense/stiff matrices	AFEM $O(N^{-1/2})$ (Lischke et al., 2018)
Walk-on-spheres	Bounded domains	Stochastic (Feynman-Kac), mesh-free	$O(1/\sqrt{M})$ MC error (Lischke et al., 2018, Valenzuela, 2022)
Deep neural networks	Bounded domains, high-dim	Stochastic representation, overcomes curse of dimensionality	$O(\varepsilon)$ $L^2$ error realized (Valenzuela, 2022)

Key implementation considerations:

• Far-field truncation, tail treatment (e.g., asymptotic or exact integration of nonlocal influences) (Huang et al., 2013, Huang et al., 2016).
• Discrete maximum principle and monotonicity for nonlinear and obstacle problems (Huang et al., 2013).
• Preconditioning by local Laplacians can improve the conditioning for Krylov solvers in higher dimensions (Minden et al., 2018).
• Efficient Gauss–Legendre-based quadrature schemes (Duffy transform) handle singular double integrals in 3D FEM (Feist et al., 2022).
• High-dimensional neural network solvers leverage stochastic Feynman–Kac representations to bypass the curse of dimensionality (Valenzuela, 2022).

6. Special Cases, Extensions, and Applications

• Explicit Action on Power Functions: Closed-form evaluation for $u_p(x) = (1 - |x|^2)_+^p$ in $\mathbb{R}^d$ in terms of hypergeometric functions enables spectral and variational analysis on spheres/balls (e.g., for computing eigenvalues) (Dyda, 2011).
• Polynomial-Growth Functions: For functions of polynomial growth at infinity, the classical PV definition diverges; an extension is defined up to a polynomial ambiguity, with optimal Schauder-type estimates and Liouville theorems in equivalence classes modulo polynomials (Dipierro et al., 2016).
• Nonlocal Discrete Models and Periodic Kernel Limit: Discrete periodic fractional Laplacians, when properly scaled as lattice spacing $h\to0$, yield continuum operators with explicit $L$-periodic Riesz kernels, relevant for anomalous diffusion and fractional quantum mechanics (Michelitsch et al., 2014).

Fractional Laplacians are central in modeling jump processes, anomalous transport, phase transitions, image processing, finance (option pricing under Lévy models), peridynamics, and geometric analysis of nonlocal curvatures (Teymurazyan, 2023, Lischke et al., 2018). Variable- and anisotropic-order extensions model complex materials with spatial or directional heterogeneity (Darve et al., 2021, Zheng et al., 2023).

7. Open Problems and Methodological Perspective

• The non-uniqueness of the fractional Laplacian in bounded domains (Riesz-integral vs. spectral vs. regional) and the matching of physical boundary data remain active areas of research (Lischke et al., 2018, Harizanov et al., 2020).
• Regularity theory, boundary Harnack inequalities, and adaptive numerical error control for solutions near the boundary depend delicately on the operator’s definition and the order $s$ (Harizanov et al., 2020, Lischke et al., 2018, Teymurazyan, 2023).
• The design of efficient fast solvers (e.g., multigrid preconditioners for dense nonlocal FEM matrices) and scalable meshfree or deep-learning based approaches for high-dimensional problems present ongoing challenges (Minden et al., 2018, Valenzuela, 2022, Cayama et al., 2022).

The fractional Laplacian’s multiple guises (Fourier, Riesz, spectral, Mellin, variable order, matrix-valued) and its intricate nonlocal character make it a unifying theme in analysis, probability, and applied mathematics (Teymurazyan, 2023, Pagnini et al., 2023, Darve et al., 2021).