Tree Capacity and Splitting Isometries for Subinvariant Kernels

Starting from a subinvariant positive definite kernel under a branching pullback, we attach to the resulting kernel tower a canonical electrical network on the word tree whose edge weights are the diagonal increments. This converts diagonal growth into effective resistance and capacity, giving explicit criteria and quantitative bounds, together with a matching upper bound under a mild level regularity condition. When the diagonal tower has finite limit at a point, we prove convergence of the full kernels and obtain an invariant completion with a minimality property. We also describe the associated RKHS splitting and a boundary martingale construction leading to weighted invariant majorants.

💡 Research Summary

The paper develops a comprehensive framework for analyzing sub‑invariant positive definite kernels by translating the kernel iteration into an electrical network on a rooted m‑ary word tree. Starting with a set X and maps φ₁,…,φ_m : X→X, the pull‑back operator L is defined by (LJ)(s,t)=∑{i=1}^m J(φ_i(s),φ_i(t)). Assuming sub‑invariance L K ≥ K for a given kernel K, the authors build a monotone tower K₀=K, K{n+1}=L K_n. The diagonal entries u_n(s)=K_n(s,s) form an increasing sequence; the one‑step increment a_s(w)=u_{|w|+1}(φ_w(s))−u_{|w|}(φ_w(s)) at a node w of the word tree is used as the conductance of each outgoing edge from w, scaled by m^{|w|+1}. This yields a canonical electrical network whose edge weights depend solely on the kernel’s diagonal growth.

Using classical tree‑network theory, the total conductance of the level‑k cutset is shown to be C_k(s)= (u_{2k+1}(s)−u_{2k}(s))/m^k. For any unit flow θ to depth N, the energy satisfies E_s(θ)≥∑{k=0}^{N-1}1/C_k(s), which translates into a lower bound on the effective resistance R_N(s)≥∑{k=0}^{N-1}(u_{2k+1}(s)−u_{2k}(s))/m^k and an upper bound on the capacity Cap_N(s)=1/R_N(s). Consequently, the series S(s)=∑{k=0}^∞ (u{2k+1}(s)−u_{2k}(s))/m^k determines whether the limiting capacity is zero (S(s)=∞) or possibly positive (S(s)<∞).

To obtain matching upper bounds, the authors impose a mild concentration condition on the level‑wise distribution of increments: the ratio of total increment to the number of non‑zero increments at level k is bounded by Λ m^{2k}. Under this hypothesis, a uniform splitting flow has energy within a factor Λ of the lower bound, yielding two‑sided estimates (1/Λ)∑{k=0}^{N-1}(u{2k+1}(s)−u_{2k}(s))/m^k ≤ R_N(s) ≤ Λ∑{k=0}^{N-1}(u{2k+1}(s)−u_{2k}(s))/m^k. Thus the diagonal tower alone controls the tree capacity, providing an “if and only if” criterion for positive limiting capacity.

The paper then focuses on points where the diagonal limit u_∞(s)=lim_{n→∞}u_n(s) is finite, defining X_fin = {s∈X : u_∞(s)<∞}. Because each increment K_{n+1}−K_n is positive definite, the finiteness of the diagonal forces uniform bounds on the off‑diagonal entries, guaranteeing pointwise convergence of the full kernels K_n(s,t) to a limit K_∞(s,t). This limit satisfies L K_∞=K_∞ and enjoys a minimality property: among all L‑invariant kernels on X_fin, K_∞ is the smallest in the Loewner order. The authors call K_∞ the invariant completion of K on X_fin.

In the reproducing kernel Hilbert space H(K_∞), the authors construct word operators S_w indexed by words w∈W_* via (S_w f)(s)=f(φ_w(s)). Sub‑invariance yields the orthogonal splitting identity ∑{|w|=n} S_w^* S_w = I for each level n, establishing a row‑isometric splitting (a Wold‑type decomposition) of H(K∞). This provides an explicit isometric embedding of the kernel dynamics into a shift‑invariant operator algebra.

The boundary theory is developed by sampling infinite words ω∈Ω={1,…,m}^ℕ. For each ω, the martingale M_n(s,t;ω)=m^n K_∞(φ_{ω|n}(s), φ_{ω|n}(t)) is shown to converge almost surely under an L² condition involving the diagonal along the sampled path: ∑{k=0}^∞ u∞(φ_{ω|k}(s))/m^k < ∞. The limit M_∞(s,t;ω) satisfies a shift‑cocycle relation M_∞(s,t;σ_i ω)=M_∞(φ_i(s), φ_i(t);ω). The diagonal boundary factors h(s;ω)=M_∞(s,s;ω) dominate all off‑diagonal values, enabling the construction of a large cone of L‑invariant positive definite kernels: J_f(s,t)=∫Ω f(ω) M∞(s,t;ω) dμ(ω), where f≥0 is any integrable weight. Constant weight recovers K_∞, while varying weights produce new invariant majorants.

Overall, the paper establishes a novel bridge between kernel dynamics, electrical network theory on trees, and operator‑theoretic structures. By encoding diagonal growth as conductances, it translates scalar information into effective resistance and capacity, yielding sharp quantitative criteria for the existence of a non‑trivial invariant kernel. The subsequent RKHS splitting and boundary martingale constructions provide powerful tools for generating and analyzing invariant majorants, opening avenues for applications in fractal analysis, multiscale signal processing, and stochastic models where sub‑invariant kernels arise.