Discrete Fractional Laplacian

• Discrete Fractional Laplacian is a nonlocal operator defined on discrete structures, generalizing the classical difference Laplacian with long-range interactions.
• It is characterized by representations via semigroup, Fourier, and convolution methods that enable rigorous spectral and probabilistic analysis.
• Applications include anomalous diffusion, numerical discretization, and spectral theory, providing practical tools for modeling discrete physical systems.

The discrete fractional Laplacian is a nonlocal operator acting on functions defined on discrete sets such as integer lattices, periodic rings, finite graphs, or meshes, and is parameterized by an order $s \in (0,2)$ (sometimes denoted $\alpha$). It generalizes the classical second-difference Laplacian by encoding algebraically decaying, long-range interactions and bridges the gap between nearest-neighbor diffusion and Lévy-type jump processes, with critical applications in anomalous transport, spectral theory, numerical analysis, and stochastic processes on discrete spaces.

1. Definitions and Fundamental Properties

On $\mathbb{Z}$, the fractional discrete Laplacian $(-\Delta)^s$ is precisely defined via several equivalent frameworks:

• Semigroup/Balakrishnan Representation:

$$(-\Delta)^s u_n = \frac{1}{\Gamma(-s)} \int_0^\infty z^{-s-1} (e^{-z\Delta} u_n - u_n) \, dz,$$

where $\Delta u_n = u_{n+1} - 2u_n + u_{n-1}$ and $e^{-z\Delta}$ denotes the discrete heat semigroup (Padgett et al., 2019, Li et al., 11 Apr 2025, Ciaurri et al., 2015).

• Spectral/Discrete Fourier Transform:

$$\widehat{(-\Delta)^s u}(\xi) = [4 \sin^2 (\xi / 2)]^s \, \hat{u}(\xi),$$

with

$$\hat{u}(\xi) = \sum_{n \in \mathbb{Z}} u_n e^{-i \xi n}, \quad \xi \in [-\pi, \pi].$$

• Convolution (Integral-Kernel) Representation:

$$(-\Delta)^s u_n = \sum_{m \in \mathbb{Z}} (u_n - u_m) K_s(n - m),$$

where the jump kernel $K_s(m)$ admits the explicit formula

$$K_s(m) = \frac{4^s \Gamma(\frac{1}{2} + s)}{\sqrt{\pi} |\Gamma(-s)|} \frac{\Gamma(|m| - s)}{\Gamma(|m| + 1 + s)}, \quad m \neq 0,$$

with asymptotic decay $K_s(m) \sim |m|^{-(1+2s)}$ for large $|m|$ (Padgett et al., 2019, Molina, 2019).

These constructions generalize to $\mathbb{Z}^d$ and finite graphs using the spectral calculus of the underlying (positive semidefinite) discrete Laplacian (Athmouni, 12 Oct 2025, Zhang et al., 2024).

On finite graphs, for Laplacian eigenpairs $-\Delta \phi_k = \lambda_k \phi_k$, the fractional power is

$$(-\Delta)^s f = \sum_{k=1}^n \lambda_k^s \langle f, \phi_k \rangle \phi_k,$$

and approaches the identity map as $s \to 0$ and the classical Laplacian as $s \to 1$ (Zhang et al., 2024).

2. Analytical Features and Probabilistic Interpretation

The operator $(-\Delta)^s$ is nonlocal, coupling every site to all others with algebraically decaying weight. For a function $f$ on $\mathbb{Z}$, $(-\Delta)^s$ corresponds to the generator of a continuous-time Markov jump process whose jump probabilities are given by $K_s(m)$, and whose associated semigroup $e^{-t(-\Delta)^s}$ admits a subordinate representation in terms of the heat kernel (Li et al., 11 Apr 2025, Kräss et al., 2022). Long-range nonlocality underlies anomalous diffusion, with transition from ballistic/normal diffusion (for $s=1$) to super-diffusive ($s \in (0,1)$) and sub-diffusive regimes ($s \in (1,2)$), as confirmed by scaling analyses of the mean-square displacement and spectral measure techniques (Padgett et al., 2019, Molina, 2019).

Analytically, the domain of $(-\Delta)^s$ in $\ell^p(\mathbb{Z})$ is characterized by summability conditions,

$\sum_m |f(n) - f(m)| K_s(n-m) < \infty,$

with precise fractional discrete Sobolev/Hölder regularity estimates and discrete mean value properties and embeddings (Ciaurri et al., 2015, Ciaurri et al., 2016, Zhang et al., 2024).

Heat kernel estimates exhibit sub-Gaussian decay, with rigorous Li-Yau and Harnack inequalities available through curvature-dimension methods for operators of the generic form

$$L u(x) = \sum_{y \in \mathbb{Z}} k(x-y) (u(y) - u(x)),$$

with $k$ symmetric, integrable, and power-law decay (Kräss et al., 2022).

3. Boundary Conditions and Extensions

Boundary conditions are implemented either by truncation (Dirichlet, $u_n\equiv 0$ outside a domain), or periodic extension (ring topology). In periodic settings, the kernel is periodized:

$$K_{A_N}^s(j) = \sum_{k \in \mathbb{Z}^d} K^s(j + k (2N+1)),$$

with the transference principle ensuring that the periodic fractional operator coincides with the restriction of the full-lattice operator followed by projection (Fernández-Bertolin et al., 2022, Michelitsch et al., 2014). On the half-lattice, the operator decomposes as the restriction of the full-lattice fractional Laplacian plus a compact boundary correction, which preserves essential spectrum and all interior threshold structures (Athmouni, 12 Oct 2025).

Notably, form-theoretic constructions permit negative fractional powers, with $(-\Delta)^{-s}$ realized via dual semigroup formulas and used as discrete fractional potentials (Ciaurri et al., 2015, Ciaurri et al., 2016).

The discrete Caffarelli-Silvestre extension—realized on diamond graphs or on the half-strip of $\mathbb{Z}_h \times (0, \infty)$—identifies $(-\Delta)^s$ as the Dirichlet-to-Neumann map of degenerate elliptic problems with appropriate vertical conductance decay (Garban, 2023, Ciaurri et al., 2016).

4. Numerical Discretization and Fast Algorithms

Numerical approaches for $(-\Delta)^s$ on uniform meshes use semigroup-based convolution formulas, quadrature approximations of singular integrals, and spectral methods that preserve the exact multiplier at the discrete Fourier level (Huang et al., 2013, Huang et al., 2016, Zhou et al., 2023, Wu et al., 2021, Cayama et al., 2022, Minden et al., 2018, Zhang et al., 2024). The discrete operator generically appears as

$$L_h u_j = \sum_{k \in \mathbb{Z}} (u_j - u_{j-k}) w_k,$$

where $w_k \sim h^{-\alpha} |k|^{-(1+\alpha)}$ for large $|k|$ (flat-tailed decay), and the stencil weights can be computed either by discrete Fourier inversion or analytic integration of interpolated kernels.

Many schemes (periodic-regularized, Grünwald-Letnikov, quadrature-based, operator-factorization) yield multilevel Toeplitz matrices, admitting $O(N\log N)$ matrix-vector multiplication via FFTs (Zhou et al., 2023, Minden et al., 2018, Wu et al., 2021). Spectral methods have exact discrete symbols and attain spectral accuracy for smooth data (Zhou et al., 2023), while quadrature-based and Lagrange-interpolation approaches yield $O(h^{3-\alpha})$ or $O(h^{4})$ convergence rates depending on the basis and regularity (Huang et al., 2013, Wu et al., 2021).

Preconditioning using the local Laplacian or multigrid/BPX-type hierarchy is essential for efficient Krylov solvers, with theoretical condition-number bounds independent of mesh size (Bærland et al., 2018, Minden et al., 2018).

5. Spectral Theory, Dynamical, and Inverse Properties

The spectrum of $(-\Delta)^s$ on infinite lattices is absolutely continuous, with plane waves as generalized eigenfunctions and symbol $[4\sin^2(\xi/2)]^s$ (Padgett et al., 2019, Molina, 2019, Athmouni, 12 Oct 2025). Random perturbations (e.g., Anderson-type Hamiltonians) induce localization transitions, interpolating between ballistic and localized regimes depending on $s$, with sub-diffusive cases ($s>1$) showing strong localization and super-diffusive ($s<1$) cases supporting extended states (Padgett et al., 2019, Molina, 2019).

On the half-lattice, compact boundary corrections do not alter the essential spectrum or the finite set of interior threshold energies. For decaying potentials, form-theoretic Mourre estimates, local Limiting Absorption Principle, and weighted propagation bounds guarantee absence of singular continuous spectrum and completeness of local wave operators (Athmouni, 12 Oct 2025).

Unique continuation properties, crucial in inverse problems, are markedly different in the discrete setting: the discrete fractional Laplacian fails global unique continuation on finite or periodic lattices (nontrivial "ghost" solutions exist) (Fernández-Bertolin et al., 2022, Kräss et al., 2022). However, continuum-like rigidity and stability can be recovered up to exponentially small corrections in the mesh size, yielding discrete Calderón-type logarithmic stability for inverse problems (Fernández-Bertolin et al., 2022).

6. Connections to Physical Models and Generalizations

The discrete fractional Laplacian arises as the generator of Lévy flights and anomalous diffusion in discrete spaces, the kinetic term in fractional quantum mechanics, and as the scaling limit of self-similar spring models with the fractional continuum limit converging to hypersingular integrals (Michelitsch et al., 2014, Michelitsch et al., 2015). Explicit finite-$N$ matrix representations can be constructed, allowing for fine control of finite-size effects and periodicity.

The operator extends to arbitrary graphs using eigenvalue decomposition, with applications to nonlinear problems such as the fractional Kazdan–Warner equation, where solutions can be obtained via variational methods or upper/lower-solution constructions and have corresponding functional inequalities (Zhang et al., 2024).

Energy identities, Sobolev and Poincaré inequalities, and Hardy–Littlewood–Sobolev embeddings hold for positive and negative powers, with discrete analogues of classical regularity, maximum principles, and mean-value properties (Ciaurri et al., 2016, Ciaurri et al., 2015, Kräss et al., 2022).

7. Summary Table: Key Discrete Fractional Laplacian Representations

Setting	Formula for $(-\Delta)^s u_n$	Reference
$\mathbb{Z}$, full	$\sum_{m\ne n} (u_n-u_m) K_s(n-m)$	(Padgett et al., 2019Ciaurri et al., 2015)
$\mathbb{Z}_h$ (mesh)	$h^{-2s}\sum_{m\ne j} (u_j-u_m)K_s(j-m)$	(Ciaurri et al., 2015Ciaurri et al., 2016)
Graphs ($G$ finite)	$$\sum_{k=1}^n \lambda_k^s \langle u,\phi_k \rangle \phi_k$$	(Zhang et al., 2024)
Periodic ring	$(u_n-u_{m}) K_{A_N}^{s}(n-m)$ (periodized)	(1412.59042202.02724)
Half-lattice	Restriction + compact boundary correction	(Athmouni, 12 Oct 2025)
Arbitrary kernel	$\sum_{y} k(x-y) (u(y)-u(x))$	(Kräss et al., 2022)

The discrete fractional Laplacian thus constitutes a rigorously defined, structurally flexible, and analytically rich operator, fundamental for modeling nonlocal dynamics and extending continuous theories to discrete settings. Its spectral, probabilistic, numerical, and inverse properties are now comprehensively understood and implemented in contemporary analysis.