Discrete Caffarelli–Silvestre Extension

• The discrete Caffarelli–Silvestre extension is a method that reformulates fractional Laplacians on graphs as local elliptic problems, establishing a Dirichlet-to-Neumann correspondence.
• It utilizes spectral calculus and the heat semigroup approach to rigorously define fractional powers and ensure regularity via sub-Gaussian heat kernel estimates and Harnack inequalities.
• The framework supports efficient computational implementations, including finite and boundary element methods, with applications spanning random interface models and fractional random walks.

The discrete Caffarelli–Silvestre extension provides a framework to realize discrete fractional Laplacians and related nonlocal operators as Dirichlet-to-Neumann maps of local, higher-dimensional elliptic problems on discrete spaces such as graphs or lattices. This approach, inspired by the foundational work of Caffarelli and Silvestre for the continuum fractional Laplacian, enables the transfer of analytical and geometric tools from local to nonlocal settings, facilitating the study of regularity, Harnack inequalities, and effective computational methods for problems involving discrete fractional powers of graph Laplacians and other non-local operators (Baudoin et al., 2024, Garban, 2023, Faustmann et al., 2023).

1. Discrete Dirichlet Forms and Sub-Gaussian Geometry

In the discrete setting, consider a countable vertex set $V$, reference measure $\mu: V \to (0, \infty)$, and a symmetric weight function $a_{xy} = a_{yx} \geq 0$, $a_{xx} = 0$. The Dirichlet form is

$$\mathcal{E}(f, g) = \frac{1}{2} \sum_{x, y \in V} a_{xy} (f(x) - f(y))(g(x) - g(y))$$

on finitely supported functions, with closure $(\mathcal{E}, \mathcal{F})$ in $\ell^2(V, \mu)$. The associated graph Laplacian is

$$L f(x) = \frac{1}{\mu(x)} \sum_{y \in V} a_{xy} (f(x) - f(y)).$$

A key geometric assumption is volume-doubling: for $C_D > 0$, all $x \in V$ and $R>0$,

$\mu(B(x, 2R)) \leq C_D\,\mu(B(x, R)),$

where $B(x, R)$ is the metric ball. For analytic control, a sub-Gaussian upper bound for the heat kernel $p_t(x, y)$ associated to the semigroup $P_t = e^{tL}$ is imposed: $$p_t(x, y) \leq C t^{-d_H/d_W} \exp\left(- c \left(\frac{d(x, y)^{d_W}}{t}\right)^{1/(d_W-1)} \right),$$ with $d_H > 0$ (Hausdorff dimension) and $d_W > 2$ (walk dimension) (Baudoin et al., 2024).

2. Discrete Fractional Laplacians

For $0 < s < 1$, the discrete fractional power $L^s$ of the graph Laplacian is defined either spectrally or via semigroups:

• Spectral calculus:

$L^s f = \int_0^\infty \lambda^s dE(\lambda)\,f,$

with domain

$$\mathcal{D}(L^s) = \left\{ f : \int_0^\infty \lambda^{2s}\,d\langle E(\lambda) f, f \rangle < \infty \right\}.$$

• Heat semigroup (Balakrishnan representation):

$$L^s f = \frac{1}{\Gamma(-s)} \int_0^\infty (P_t f - f) t^{-1-s} dt, \qquad \Gamma(-s) = -\frac{\Gamma(1-s)}{s}.$$

This generalizes the continuous theory and provides a robust framework for nonlocal operators on discrete spaces (Baudoin et al., 2024).

3. Extension Problem and Boundary Representation

The discrete Caffarelli–Silvestre extension seeks $U : V \times [0, \infty) \to \mathbb{R}$ satisfying

$$-\partial_y (y^a \partial_y U(x, y)) + y^a L_x U(x, y) = 0, \qquad U(x, 0) = f(x),$$

with weight exponent $a = 1-2s \in (-1, 1)$. There is a unique bounded solution given by the discrete Poisson integral,

$U(x, y) = \int_V P_y(x, z) f(z) d\mu(z),$

where $P_y(x, z)$ admits the integral representation

$$P_y(x, z) = \frac{y^{2s}}{4^s \Gamma(s)} \int_0^\infty e^{-y^2/(4t)} p_t(x, z) t^{-1-s} dt.$$

Critically, the Dirichlet-to-Neumann map recovers the fractional power: $$-\lim_{y \to 0^+} y^a \partial_y U(x, y) = d_s L^s f(x), \qquad d_s = \frac{2^{2s-1}\Gamma(s)}{\Gamma(1-s)}.$$ This establishes the equivalence between nonlocal equations on $V$ and local (degenerate) elliptic equations on the cylinder $V \times (0, \infty)$ (Baudoin et al., 2024, Garban, 2023).

4. Discrete Models: Lattices and Half-Grids

For the canonical case of $\mathbb{Z}$, the “half-grid” construction gives a natural graph Laplacian on $V = \mathbb{Z} \times \mathbb{N}$, with conductances chosen to enforce the correct regenerative (Markov) structure and matching to the desired discrete fractional operator at the boundary:

• Horizontal edges: conductance $1/2$.
• Vertical edges at height $y$:

$$a(y) = \begin{cases} 1, & y = 0 \ \frac{1}{2} - \frac{s}{2(1+y)} \geq \frac{1}{4}, & y \geq 1 \end{cases}$$

The discrete Dirichlet-to-Neumann map is

$(\Lambda_s u)(i) = c_s a(0) (U(i, 1) - U(i, 0)),$

with $\Lambda_s u = (-\Delta)^J u$ for a choice of weights $J(r) \sim r^{-(1+2s)}$, ensuring the extension recovers the intended long-range discrete Laplacian (Garban, 2023).

The well-posedness via convex minimization of the energy, energy identity between extension and original fractional energy, and regularity (decay of $U(i, y)$ as $y \to 0$ and discrete-harmonicity for $y \geq 1$) are rigorously established. This supports applications to statistical mechanics and random interface models, such as the discrete Gaussian chain and higher-dimensional analogues (Garban, 2023).

5. Regularity: Hölder Continuity and Harnack Inequality

The sub-Gaussian heat kernel estimates on $V$ are sufficient to establish regularity properties for solutions to $L^s f = 0$:

• Hölder continuity:

$$|f(x) - f(x')| \leq C \left( \frac{d(x, x')}{R} \right)^\alpha \operatorname{osc}_{B(x_0, R)} f,$$

for some $\alpha \in (0, 1)$ depending on $s$, $d_H$, and $d_W$.

• Global elliptic Harnack inequality:

$\sup_{B(x_0, R)} f \leq C \inf_{B(x_0, R)} f,$

for non-negative solutions. The proof utilizes the “lift-and-slice” method: regularity and parabolic Harnack inequalities are proved for the extended caloric problem, and then restricting to the boundary $y = 0$ yields estimates for the original nonlocal equation. The structure and constants in these estimates are determined by the metric, measure, and heat kernel geometry of $V$ (Baudoin et al., 2024).

6. Computational Realizations and Numerical Methods

Finite element and boundary element methods have been adapted to the Caffarelli–Silvestre extension to produce fully discrete schemes for fractional diffusion equations. By diagonalizing in the extended dimension and coupling local solvers in the spatial variable via symmetric saddle-point linear systems, these numerical schemes achieve provable a-priori error bounds: $$\|u - u_h\|_{H^s(\mathbb{R}^d)} \leq C h_\Omega \|f\|_{H^1(\Omega)},$$ uniformly in $s$, with exponential decay in the truncation error of the extended variable discretization and algebraic convergence dictated by the mesh size and the polynomial degrees chosen in the discretization (Faustmann et al., 2023). These approaches benefit directly from the local realization of the discrete fractional operator.

7. Applications and Extensions

The discrete Caffarelli–Silvestre extension is instrumental in the study of random interface models, Gaussian chains, and fractional random walks. The extension provides not only analytical tools for establishing large-scale invariance principles—such as the emergence of fractional Brownian motions from random walks with long-range steps—but also enables the transfer of discrete energy methods, coupling arguments, and regularity theory to nonlocal operators on varied combinatorial and metric structures (Garban, 2023, Baudoin et al., 2024).

Generalizations encompass higher-dimensional half-spaces, random conductance models, fractional parabolic equations, and the analysis of nonlinear lattice models with fractional coupling. The technique facilitates both theoretical advances in regularity and probabilistic scaling limits, and the development of efficient solvers for large-scale discrete fractional models in applied settings (Garban, 2023, Faustmann et al., 2023).

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Key References

Paper title/subject	arXiv ID	Main focus
Extension method in Dirichlet spaces with sub-Gaussian estimates...	(Baudoin et al., 2024)	General discrete extension for Dirichlet forms, regularity, Harnack
Invisibility of the integers for the discrete Gaussian chain via...	(Garban, 2023)	Discrete CS extension for 1D/2D models, energy identity, scaling limits
FEM-BEM coupling in Fractional Diffusion	(Faustmann et al., 2023)	Fully discrete schemes, variational formulation, numerical analysis