A new graph-directed construction of nonlocal energies on the unit interval

TL;DR

The paper develops a novel graph-directed framework to construct nonlocal energies on the unit interval by taking limits of discrete energies on complete electrical networks built from dyadic dyads. It shows that, under kernel bounds comparable to a fractional discrete Laplacian with exponent $s\in(0,1)$, the discrete energies Mosco-converge to a limiting nonlocal energy that is equivalent to the fractional Gagliardo seminorm on appropriate domains, with explicit results for the physical spaces $[0,1/i]\cup[1-1/i,1]$ and a detailed treatment of the $i\to\infty$ limit. The approach unifies two limit constructions (finite-energy and density-based) and establishes energy equivalence across the two, leveraging extensions of jump kernels and Γ-convergence arguments. This provides a robust analytic route to nonlocal Dirichlet forms on $[0,1]$ directly from discrete graph models, with potential numerical and fractal-geometry applications. The framework highlights the role of complete networks in encoding nonlocal interactions and connects network resistance concepts to Sobolev-type energies, offering a pathway to new nonlocal diffusion models on continua.

Abstract

We present an analytic construction of nonlocal energies on the unit interval. The energies are defined using a new graph-directed construction of discrete energies on dyadic approximations of the interval. When the discrete jump kernels are comparable to the kernel of the fractional discrete Laplacian, we prove that the discrete energies Mosco converge and the limiting energy is equivalent to the fractional Gagliardo seminorm.

Figures (9)

• Figure 1: Electrical networks (also called finite graph approximations)
• Figure 2: Directed graph $(\mathbb{N},E)$ in the index space
• Figure 3: Discrete approximations $V_i^{(n)}$ of the physical spaces $\overline{V}_i^*$
• Figure 4: Electrical networks
• Figure 5: Renormalization of resistances from $(V_i^{(1)}, W_i^{(1)})$ to $(V_i^{(0)}, W_i^{(0)})$

Theorems & Definitions (87)

• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• Remark 2.5
• Definition 3.1
• Remark 3.2
• Lemma 3.3
• proof