Anisotropic Fractional Discrete Laplacians

• Anisotropic fractional discrete Laplacians are nonlocal operators defined on lattices and graphs, using fractional powers and direction-dependent scaling to generalize classical Laplacians.
• They are discretized via spectral, quadrature, and energy-based methods that accurately capture singular kernel behavior and anisotropic effects.
• Applications span quantum physics, control theory, and stochastic processes, with robust theoretical and numerical frameworks addressing challenges in regularity and boundary conditions.

Anisotropic fractional discrete Laplacians are nonlocal difference operators defined on discrete structures—such as lattices or graphs—where the nonlocality and scaling behavior vary with direction. These operators generalize the classical Laplacian and its discrete counterparts by introducing fractional powers and anisotropic exponents along coordinate axes or graph directions. They appear in numerical analysis, quantum physics, control theory, probability, and applied mathematics as natural tools for modeling anomalous and directionally-dependent diffusion, nonlocal interactions, and transport phenomena.

1. Foundational Definitions and Spectral Representations

The anisotropic fractional discrete Laplacian is best formalized through spectral calculus on product discrete spaces, such as the $d$-dimensional integer lattice $\mathbb{Z}^d$. Given an exponent vector $$\vec{r} = (r_1, ..., r_d) \in \mathbb{R}^d\setminus\{0\}$$, the operator is defined as

$$\Delta^{\vec{r}}_{\mathbb{Z}^d} = \sum_{j=1}^d \left(\mathrm{id}\right)^{\otimes(j-1)} \otimes \Delta^{r_j}_{\mathbb{Z}} \otimes \left(\mathrm{id}\right)^{\otimes(d-j)},$$

where each $\Delta^{r_j}_{\mathbb{Z}}$ is the fractional power of the standard discrete Laplacian on $\ell^2(\mathbb{Z})$. Under the discrete Fourier transform, the action is diagonalized: $$\mathcal{F}(\Delta^{r_j}_{\mathbb{Z}}f)(\theta) = [2(1 - \cos \theta)]^{r_j} \mathcal{F} f(\theta).$$ The spectrum is the range of the symbol $$\theta \mapsto \sum_{j=1}^d [2(1 - \cos\theta_j)]^{r_j}$$ (Athmouni, 7 Sep 2025). Fractional powers with positive exponents yield bounded operators, while negative exponents correspond to fractional inverses with more restrictive domains.

In finite graph settings, the discrete fractional Laplacian for exponent $s$ is defined via the graph Laplacian spectrum: $$(-\Delta)^s u(x) = \sum_{i=1}^n \lambda_i^s \langle u, \varphi_i \rangle \varphi_i(x),$$ where $(\lambda_i, \varphi_i)$ are Laplacian eigenpairs (Zhang et al., 29 Mar 2024). This spectral description is key for both analytic theory and computational implementation.

2. Discretization Strategies: Quadrature, Spectral, and Energy-based Methods

A variety of numerical schemes have been established for discretizing fractional Laplacians on regular grids, graphs, and lattices:

• Spectral methods: Employ direct diagonalization and apply fractional powers to eigenvalues (Zhang et al., 29 Mar 2024).
• Finite difference/quadrature methods: Discretize the hypersingular integral kernel using suitable convolution weights $w_k$ that reflect the desired fractional and anisotropic scaling. In higher dimensions, the weights become direction-dependent, encoding anisotropy (Huang et al., 2016, Huang et al., 2013).
• Energy/variational approaches: For fractional Laplacians defined via quadratic forms or minimization of weighted Dirichlet energies (such as the Caffarelli–Silvestre or Stinga–Torrea extension frameworks), discrete analogs use anisotropically weighted difference quotients on boxes, rectangles, or graphs (Chaker et al., 2021, Musina et al., 2014).

Weights are often chosen so that for large $k$, $w_k \sim h^{-\alpha} / k^{1+\alpha}$, exhibiting nonlocality and slow, algebraic “flat tail” decay. For anisotropic problems, the discrete kernel or metric is replaced by a function that scales differently in each direction.

3. Variational Principles and Sobolev-type Inequalities

Analysis of anisotropic fractional discrete Laplacians often employs energy-based techniques:

• Quadratic form inequalities: For classic fractional Laplacians, the Navier and Dirichlet realizations for $s \in (0,1)$ satisfy positivity-preserving and positive-definite difference properties, implying ordering of spectral values and strict inequalities between quadratic forms (Musina et al., 2013, Musina et al., 2014, Nazarov, 2021).
• Sobolev and Besov space embeddings: Discrete operators equipped with anisotropic kernels give rise to anisotropic Besov and Sobolev spaces, characterized via semigroup or difference-based seminorms. Threshold phenomena for critical exponents $\alpha_p^\#$ delineate the existence of nonconstant functions and regulate regularity (Ruiz et al., 2019).

A typical anisotropic Sobolev-type inequality for mixed-order orthotropic $p$-Laplacians states

$$\|u\|_{L^{p_*}(\mathbb{R}^n)}^p \le C \iint_{\mathbb{R}^n \times \mathbb{R}^n} |u(x) - u(y)|^p \, \mu_{\mathrm{axes}}(x, dy)\, dx,$$

with the exponent governed by the harmonic mean of the orders and anisotropic rectangle domains (Chaker et al., 2021).

4. Fundamental Properties: Nonlocality, Regularity, and Boundary Effects

Anisotropic fractional discrete Laplacians are fundamentally nonlocal—individual entries involve contributions from distant lattice points or graph vertices. Analytical and numerical challenges stem from:

• Singular kernel behavior: The convolutional weights encode strong singularity near the origin, demanding careful quadrature or singularity subtraction for accurate discretization (Minden et al., 2018).
• Nonlocal boundary influence: In Riesz-type definitions, values outside a bounded domain must be prescribed (“volume constraint”), whereas spectral definitions require only boundary conditions (Lischke et al., 2018).
• Regularity theory: Robust Harnack and local Hölder estimates demonstrate that solutions possess interior regularity, even in highly anisotropic contexts and for nonlinear, degenerate equations (Chaker et al., 2021).
• Positivity and maximum principles: Certain discretizations maintain positivity and obey discrete maximum principles, ensuring stability and monotonicity of numerical schemes (Huang et al., 2013).

5. Stochastic, Dynamical, and Transport Interpretations

Fractional discrete Laplacians, especially in the anisotropic case, admit stochastic process interpretations:

• Lévy processes: Riesz-type fractional operators are generators of symmetric (isotropic) or anisotropic $\alpha$-stable processes. Biasing jump direction via non-uniform measures yields anisotropy (Lischke et al., 2018).
• Spectral transport: On lattices, propagation and minimal velocity estimates derived via Mourre theory quantify ballistic transport and local decay—key for time-dependent models (Athmouni, 7 Sep 2025).
• Wave and scattering theory: On hypercubic lattices, anisotropic fractional Laplacians admit comprehensive scattering frameworks, including construction of the stationary scattering matrix $S(\lambda)$, the optical theorem, and the Birman–Kreĭn formula $\det S(\lambda) = \exp(-2\pi i\,\xi(\lambda))$, connecting transport, resonances, and spectral shift functions (Athmouni, 7 Sep 2025).

6. Applications in Nonlinear and Fully Nonlocal Problems

Anisotropic fractional discrete Laplacians are prominent in:

• Stochastic control and games: Numerical schemes discretizing fractional Hamilton–Jacobi–BeLLMan and Isaacs equations achieve uniform second-order accuracy, monotonicity, and $L^\infty$ stability independent of the fractional order (Chowdhury et al., 18 Jan 2024).
• Kazdan–Warner-type equations on graphs: The spectral fractional Laplacian underpins existence results and variational analyses for nonlinear problems. Its asymptotic behavior as $s\to1$ or $s\to0$ interpolates between classical Laplacians and identity maps (Zhang et al., 29 Mar 2024).
• Anomalous transport and diffusion: Models in porous media, finance, and physics exploit the directionally-biased nonlocal interactions encoded by anisotropic kernels, impacting heat kernel estimates and isoperimetric inequalities (Ruiz et al., 2019, Lischke et al., 2018).
• Numerical solvers: FFT-accelerated convolutional schemes, preconditioned Krylov methods, adaptive finite elements, and Monte Carlo techniques—such as walk-on-spheres algorithms—provide practical high-dimensional solvers for fractional and anisotropic diffusion (Minden et al., 2018, Lischke et al., 2018).

7. Challenges and Future Directions

Open questions include:

• Accurate computation of multidimensional, anisotropic weights: Oscillatory quadrature for highly nonlocal, direction-dependent kernels remains computationally demanding (Huang et al., 2016).
• Boundary condition formulation for nonlocal operators: Nonlocal Neumann and Robin conditions, especially for Riesz-type and horizon-truncated operators, lack consensus (Lischke et al., 2018).
• High-dimensional scaling and preconditioning: Efficient solvers and well-conditioned discretizations in domains with pronounced anisotropy are in active development, with directions such as RBF collocation and spectral element methods showing promise (Lischke et al., 2018, Minden et al., 2018).
• Analysis across limits of the fractional order: Smooth interpolation between local and nonlocal phenomena, phase transitions, and variable interaction range, as described for spectral families parametrized by $\vec{r}$, require further theoretical and numerical elucidation (Athmouni, 7 Sep 2025).

In summary, anisotropic fractional discrete Laplacians encapsulate a spectrum of nonlocal operators with broad applicability, robust theoretical underpinnings, and rich connections to spectral theory, probability, numerical analysis, and applied sciences. The current literature provides comprehensive frameworks for their definition, discretization, regularity properties, and dynamical interpretations, with ongoing research addressing mathematical, computational, and modeling challenges in anisotropic, high-dimensional, and nonlinear contexts.