Fractional Laplacian

📐 Definition

Let $0 < \alpha \le 2$ . The fractional Laplacian $(-\Delta)^{\alpha/2}$ is defined, for sufficiently regular functions $f$ , by its action in Fourier space:

\mathcal{F}\!\left[(-\Delta)^{\alpha/2} f\right](\xi) = |\xi|^{\alpha}\,\hat{f}(\xi)

Equivalently, it may be defined as a singular integral:

(-\Delta)^{\alpha/2} f(x) = C_{d,\alpha} \operatorname{P.V.} \int_{\mathbb{R}^d} \frac{f(x) - f(y)}{|x-y|^{d+\alpha}}\,dy

where $\operatorname{P.V.}$ denotes the Cauchy principal value and $C_{d,\alpha}$ is a normalization constant depending on the dimension $d$ and order $\alpha$ .

Domain and Codomain

The fractional Laplacian acts on functions defined on $\mathbb{R}^d$ . It is well-defined on Schwartz functions and extends to Sobolev spaces and tempered distributions. The codomain consists of real- or complex-valued functions on $\mathbb{R}^d$ .

⚙️ Key Properties

Reduction to the Classical Laplacian

For $\alpha = 2$ ,

(-\Delta)^{1} = -\Delta

recovering the standard Laplace operator.

Linearity

The fractional Laplacian is linear:

(-\Delta)^{\alpha/2}(af + bg) = a\,(-\Delta)^{\alpha/2}f + b\,(-\Delta)^{\alpha/2}g

for scalars $a,b$ .

Nonlocality

The value of $(-\Delta)^{\alpha/2} f(x)$ depends on the values of $f$ on all of $\mathbb{R}^d$ . This nonlocal character distinguishes the operator from integer-order differential operators.

Scaling Behavior

Under spatial scaling $f_\lambda(x) = f(\lambda x)$ ,

(-\Delta)^{\alpha/2} f_\lambda(x) = \lambda^{\alpha} \bigl[(-\Delta)^{\alpha/2} f\bigr](\lambda x)

This homogeneity reflects the order $\alpha$ of the operator.

Energy Form

For suitable functions $f$ ,

\int_{\mathbb{R}^d} f(x)\,(-\Delta)^{\alpha/2} f(x)\,dx \sim \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|f(x)-f(y)|^2}{|x-y|^{d+\alpha}}\,dx\,dy

This quadratic form underlies variational formulations of fractional PDEs.

🎯 Special Cases

One-Dimensional Case

In one dimension, the fractional Laplacian reduces to the Riesz fractional derivative.

Stable Lévy Processes

The generator of a symmetric $\alpha$ -stable Lévy process is $(-\Delta)^{\alpha/2}$ , linking the operator to probability theory.

The fractional Laplacian is closely related to:

the classical Laplacian,
Riesz fractional derivatives,
Fourier multipliers,
nonlocal diffusion operators.

It plays a central role in harmonic analysis and fractional PDE theory.

Usage in Oakfield

Oakfield implements a spectral fractional Laplacian as part of its diffusion operator family:

linear_dissipative operator: computes a 1D wavenumber grid from field length and spacing, forms $\lambda(k)=-|k|^\alpha$ , and applies exponential decay in frequency space with factors like $\exp(\nu\lambda(k)\,dt)=\exp(-\nu|k|^\alpha\,dt)$ .
FFT-based pipeline: the implementation follows an explicit “copy → FFT → multiply per-bin → inverse FFT → copy back” pattern with per-operator FFTPlan ownership.

Current spectral diffusion assumes a flattened 1D contiguous field when constructing $k$ (multi-D stride-aware spectral operators are not yet the default path).

Historical Foundations

📜 Origin

Marcel Riesz introduced fractional powers of the Laplacian while studying potential theory and harmonic analysis.

🔬 Analytical Development

The operator was later formalized through singular integrals and Fourier analysis, becoming a standard tool in modern PDE theory.

🌍 Modern Perspective

The fractional Laplacian is now a canonical example of a nonlocal operator, appearing in analysis, probability, physics, and numerical computation.

📚 References

M. Riesz, Intégrales de Riesz et potentiels
L. Caffarelli & L. Silvestre, An Extension Problem Related to the Fractional Laplacian
NIST Digital Library of Mathematical Functions (DLMF), §1.14