Discrete Fractional Laplacians

• Discrete Fractional Laplacians are nonlocal operators defined on lattices, graphs, and grids that extend classical Laplacians to fractional powers.
• They are constructed via spectral, semigroup, and explicit kernel methods, enabling precise modeling of long-range interactions and anomalous diffusion.
• Their applications include numerical simulation of fractional PDEs, spectral analysis on discrete spaces, and modeling random walks with heavy-tailed jump distributions.

The discrete fractional Laplacian is a nonlocal operator defined on discrete structures such as lattices, uniform grids, and finite or infinite graphs, which generalizes the classical discrete Laplacian by allowing fractional, real-valued powers. These operators play a central role in the paper of anomalous diffusion, nonlocal partial difference equations, random walks with heavy-tailed jump distributions, numerical schemes for fractional PDEs, and spectral analysis on discrete spaces. The discrete fractional Laplacian acts as the generator of long-range interactions and stable -- often anisotropic -- jump-type processes in the discrete setting, mirroring key features of its continuous analogue.

1. Definitions and Core Constructions

There are several rigorous approaches to defining the discrete fractional Laplacian, all of which extend the classical, local difference operator to realize non-integer powers and nonlocality.

Spectral (Semigroup) Definition on Lattices:

For the classical one-dimensional discrete Laplacian on a grid $\mathbb{Z}_h$,

$$( -\Delta_h u )_j = -\frac{1}{h^2}(u_{j+1} - 2u_j + u_{j-1}),$$

the fractional power $(-\Delta_h)^s$ for $0 < s < 1$ is defined via the semigroup formula:

$$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty \left( e^{t\Delta_h} u_j - u_j \right) \frac{dt}{t^{1+s}}.$$

In multidimensions, the operator’s kernel decays as $|\mathbf{m}|^{-N-2s}$, with similar convolution form and semigroup representation (Ciaurri et al., 2015, Ciaurri, 2023).

Explicit Kernel Representation:

This leads, for functions on the lattice, to pointwise nonlocal formulas such as

$$(-\Delta_h)^s u_j = \sum_{m \in \mathbb{Z} \setminus \{0\}} (u_j - u_{j-m}) K_s(m), \qquad K_s(m) = \frac{4^s\,\Gamma(\tfrac{1}{2} + s)\,\Gamma(|m|-s)}{\sqrt{\pi}|\Gamma(-s)|\,\Gamma(|m|+1+s)}.$$

This kernel decays asymptotically like $|m|^{-1-2s}$ and is symmetric (Ciaurri et al., 2015, Jones et al., 2021).

Spectral Definition on Finite Graphs:

For a finite graph $G = (V, E, \mu, w)$ with discrete Laplacian $-\Delta$ and eigenpairs $(\lambda_i, \phi_i)$,

$$(-\Delta)^s u(x) = \sum_{i=1}^n \lambda_i^s \phi_i(x) (u, \phi_i),$$

where $u \in C(V)$, and the action is diagonal in the eigenbasis. This definition extends to arbitrary $s > 0$ and converges to the classical Laplacian as $s \to 1$, and to the identity (modulo the kernel) as $s \to 0$ (Zhang et al., 29 Mar 2024, Zhang et al., 6 Aug 2024).

Heat Semigroup Formulation on Graphs:

On locally finite graphs, the operator is defined via the heat kernel $p(t,x,y)$:

$$(-\Delta)^s u(x) = \frac{1}{\Gamma(1-s)} \int_0^\infty [u(x) - e^{t\Delta}u(x)] t^{-1-s} dt = \sum_{y \in V, y \neq x} W_s(x, y)[u(x) - u(y)],$$

with $W_s(x, y)$ obtained by integrating the heat kernel against $t^{-1-s}$ (Zhang et al., 6 Aug 2024).

Functional Calculus / Series Representations:

For integer or non-integer order $s > 0$, the operator can be written as

$$(-\Delta)^s u(n) = \sum_{k \in \mathbb{Z}} K_s(k)[u(n) - u(n-k)],$$

with $K_s(k)$ defined via Gamma functions, interpolating the integer-power case by analytic continuation (Jones et al., 2021).

Finite Difference and Quadrature Schemes:

Fractional discrete Laplacians can be constructed by discretizing singular integral representations:

$$(-\Delta)^{\alpha/2} u(x) = C_{n,\alpha} \int_{\mathbb{R}^n} \frac{u(x) - u(y)}{|x-y|^{n+\alpha}} dy$$

with careful split into near-field (approximated by finite differences) and far-field (by numerical quadrature or asymptotic expansions) yielding positive, summable weights in discrete convolution operators (Huang et al., 2013, Huang et al., 2016).

2. Analytical and Spectral Properties

Nonlocality, Decay, and Regularity

• Nonlocality: The value at each site depends on a weighted sum or integral over all other sites, with algebraically decaying weights. On lattices, the kernel decays like $|m|^{-1-2s}$ (1d) or $|m|^{-N-2s}$ ($N$-dimensions).
• Spectral Interpolation: As $s \to 0$, $(-\Delta_h)^s \to$ identity; as $s \to 1$, recovers the standard discrete Laplacian. For $s \to m$ integer, the series converges to the corresponding local finite-difference operator (Jones et al., 2021).
• Regularity: The mapping properties of discrete fractional Laplacians mirror the continuous case, leading to a loss of $2s$ in discrete Hölder (or Sobolev) regularity. Discrete Schauder and Sobolev embeddings are established (Ciaurri et al., 2016, Ciaurri et al., 2015).

Mean-Value Properties and Probabilistic Interpretation

Discrete $s$-harmonic functions (i.e., $(-\Delta_h)^s u = 0$) satisfy a nonlocal mean value property: values are averages weighted by the kernel, generalizing both local and nonlocal jump processes (discrete stable Lévy flights) (Ciaurri et al., 2016, Ciaurri, 2023).

Pointwise Inequalities and Maximum Principles

Convexity-based pointwise inequalities, such as

$$(-\Delta)^s(\varphi(f))(x) \leq \varphi'(f(x)) (-\Delta)^s f(x)$$

hold in the discrete setting when the operator has a representation via positive, mass-preserving kernels. Maximum principles and discrete energy inequalities extend naturally (Cordoba et al., 2015, Huang et al., 2013).

3. Numerical Approximation and Quadrature Schemes

Finite-Difference Quadrature Approximations

Fractional Laplacians on uniform grids can be discretized by splitting the singular integral into singular (local, small $|y|$) and tail parts. Piecewise polynomial interpolation (tent or quadratic functions) is used in the tail, leading to errors of order $O(h^{k+1-\alpha})$ for degree $k$ interpolation, and positive convolution weights ensuring monotonicity and $\ell^\infty$ stability (Huang et al., 2013, Huang et al., 2016).

Symbol-Based and Spectral Methods

Convolution weights can be defined so that in frequency space the discrete symbol approximates $|\xi|^\alpha$. Spectral weights provide high (even spectral) accuracy for smooth functions, but may lose positivity and are sometimes oscillatory for $\alpha > 1$. Regularized ("periodic") and Grünwald–Letnikov weights provide positivity with first/second order consistency (Huang et al., 2016).

Far-Field Treatment and Nonlocal Boundary Conditions

Since the discrete operators are fully nonlocal, treatment of boundary or truncation errors is essential. Far-field contributions may be estimated via asymptotic expansions or using known decay (e.g., algebraic tails) in the solution, with corrections computed analytically or by numerical quadrature (Huang et al., 2013, Huang et al., 2016, Lischke et al., 2018).

4. Functional Analysis, Harmonic Analysis, and Generalizations

Harmonic Analysis and Riesz Transforms

Fractional powers on multidimensional discrete lattices, $(-A_N)^s$, are defined via the heat semigroup (or functional calculus on the Fourier side) and play a foundational role in the paper of discrete Riesz transforms and fractional integrals. The associated kernels exhibit Calderón–Zygmund-type estimates, and $g_k$-square functions are used to probe regularity and boundedness on weighted $\ell^p$ spaces (Ciaurri, 2023).

Discrete Fractional Operators in General Frameworks

Unified approaches employ generalized fractional difference (delta and nabla) operators, discrete Riemann-Liouville and Caputo types, and convolution sum formulas with prescribed kernel functions. This framework underpins stability and inversion properties and enables the modeling of anomalous diffusion, memory effects in discrete systems, and connections to fractional Laplacians (Ferreira, 2021).

Measure-Geometric and Non-Uniform Grids

For non-uniform discrete distributions, measure-geometric Laplacians are constructed using matrix representations of the derivative, yielding self-adjoint Laplacians whose powers and spectral properties are readily computed. In the uniform case, this recovers the standard circulant/tridiagonal Laplacian, and fractionalization is achieved via spectral functional calculus (Kesseböhmer et al., 2017).

5. Extension to General Graphs and Spectral Analysis

Fractional Laplacians on Graphs

On finite and infinite graphs, the operator $(-\Delta)^s$ is defined via the spectral theorem,

$(-\Delta)^s \phi_i = \lambda_i^s \phi_i,$

and/or via the heat semigroup,

$$(-\Delta)^s u(x) = \frac{1}{\Gamma(1-s)} \int_{0}^{\infty} [u(x) - \sum_{y \in V} p(t,x,y)u(y)\mu(y)] t^{-1-s} dt,$$

leading to a nonlocal, weighted summation form with explicit kernels $W_s(x,y)$ involving heat kernels (Zhang et al., 29 Mar 2024, Zhang et al., 6 Aug 2024).

Variational and PDE Theory

Fractional Sobolev spaces $W^{s,2}(V)$ are defined on graphs with norms that involve the $L^2$ term and a double sum analogous to the Gagliardo seminorm. These underlie the direct method of the calculus of variations, ground state solutions, and multiplicity results for discrete fractional Schrödinger and Kazdan–Warner equations using the mountain-pass theorem and Nehari manifold (Zhang et al., 29 Mar 2024, Zhang et al., 6 Aug 2024).

Operator Compressing and Boundary Corrections

On domains with boundaries (e.g., half-lattices), the discrete fractional Laplacian is compressed from the full lattice and augmented by a boundary correction term; this correction is relatively compact. The essential spectrum and interior threshold structure are therefore inherited from the infinite setting, and Mourre theory yields a Limiting Absorption Principle, propagation (transport) bounds, and completeness of scattering theory (Athmouni, 12 Oct 2025).

Anisotropic and Negative Fractional Orders

Anisotropic fractional Laplacians (different orders in each coordinate of $\mathbb{Z}^d$) and negative fractional powers (inverse-type nonlocal operators) are realized via functional calculus and explicit shift-operator expansions. The spectral and dynamical analysis relies on commutator techniques (Mourre estimate), yielding absence of singular spectrum, completeness of wave operators, and explicit scattering theory on the discrete structure, including the Birman–Krein formula relating the scattering matrix to spectral shift (Athmouni, 7 Sep 2025, Athmouni, 12 Oct 2025).

6. Applications, Limit Behavior, and Further Directions

Convergence and Approximation

Discrete fractional Laplacians converge to their continuum counterparts as the mesh step $h\to 0$ in the sense of operator norms or under pointwise error estimates, provided suitable regularity. This supports rigorous discretization of fractional elliptic and time-dependent PDEs for applications in anomalous transport, finance, and materials science (Ciaurri et al., 2015, Ciaurri et al., 2016, Lischke et al., 2018, Chowdhury et al., 18 Jan 2024).

Fully Nonlinear and Viscosity Solution Schemes

Second-order accurate discretizations extend to fully nonlinear equations (e.g., HJB, Isaacs equations) in viscosity solution frameworks. Monotonicity, $L^\infty$-stability, comparison principles, and convergence (via the half-relaxed limits technique) are established for wide classes of fully nonlinear, nonlocal discrete equations (Chowdhury et al., 18 Jan 2024).

Physical and Stochastic Interpretations

Discrete fractional Laplacians arise as generators of random walks with stable jump distributions on lattices or graphs. They model nonlocal processes in discrete media, serve as operators in lattice quantum mechanics, and underlie stochastic processes on networks. Spectral and transport properties govern relaxation, propagation, and localization phenomena in disordered or confined systems (Garbaczewski et al., 2018, Athmouni, 7 Sep 2025).

Operator Inequalities and Discrete Maximum Principles

Positive kernel representations allow transfer of integral inequalities and energy principles from continuum to discrete settings, underpinning stability and regularity of solutions. Discrete Hardy-Littlewood-Sobolev inequalities, Sobolev embeddings, Poincaré inequality analogs, and monotonicity of discrete schemes ensure robustness and fidelity of numerical simulation (Ciaurri et al., 2016, Cordoba et al., 2015).

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Summary Table: Core Representations for Discrete Fractional Laplacians

Framework	Core Formula / Representation	Key Features
Lattice Semigroup	$$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty [e^{t\Delta_h}u_j - u_j] \frac{dt}{t^{1+s}}$$	Convolution kernel; explicit decay; extension to $s>1$
Spectral on Graphs	$$(-\Delta)^s u(x) = \sum_i \lambda_i^s \phi_i(x) (u, \phi_i)$$	Eigen-decomposition; convergence as $s\to1$ or $s\to0$
Explicit Kernel	$(-\Delta)^s u(n) = \sum_k K_s(k)[u(n) - u(n-k)]$	Gamma function kernel; analytic continuation
Finite Difference-Q	$$(-\Delta_h)^s u_j = \sum_{m\neq 0} (u_j - u_{j-m}) K_s(m)$$, $K_s(m) \sim |m|^{-1-2s}$	Positive weights; high-order convergence
Heat Semigroup (Graph)	$$(-\Delta)^s u(x) = \frac{1}{\Gamma(1-s)} \int_0^\infty [u(x) - e^{t\Delta} u(x)] t^{-1-s} dt$$	Nonlocality via heat kernel

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Discrete fractional Laplacians form an essential class of nonlocal operators at the intersection of discrete analysis, spectral theory, numerical PDEs, and stochastic processes. Their theory extends the reach of fractional calculus into lattice models, discrete networks, and graph-based data, providing rigorous, explicit, and computationally robust tools for both analysis and simulation of nonlocal dynamics in discrete settings.