Subinvariant kernel dynamics

We study positive definite kernels pulled back along a finite family of self-maps under a subinvariance inequality for the associated branching operator. Iteration produces an increasing kernel tower with defect kernels. Under diagonal boundedness, the tower has a smallest invariant majorant, with a canonical defect space realization and an explicit diagonal harmonic envelope governing finiteness versus blow-up. We also give probabilistic and boundary representations: a Gaussian martingale model whose quadratic variation is the defect sequence, and canonical Doob path measures with a boundary feature model for the normalized defects.

💡 Research Summary

The paper investigates a purely combinatorial framework in which a positive‑definite kernel K on an arbitrary set X is repeatedly pulled back along a finite family of self‑maps ϕ₁,…,ϕₘ. The associated “branching” operator L acts on kernels by (LJ)(s,t)=∑_{i=1}^{m}J(ϕ_i(s),ϕ_i(t)). The central hypothesis is the sub‑invariance inequality LK ≥ K, which is equivalent to the defect kernel D:=LK−K being positive‑definite. Starting from K₀:=K and defining Kₙ₊₁:=L Kₙ, one obtains an increasing tower of kernels K₀≤K₁≤… together with defect increments Dₙ:=Kₙ₊₁−Kₙ = LⁿD.

Under the modest diagonal boundedness condition supₙ Kₙ(s,s)<∞ for every s∈X, the authors prove that the pointwise limit K∞(s,t)=limₙKₙ(s,t) exists and defines a positive‑definite kernel. Moreover K∞ dominates K, satisfies LK∞=K∞, and is the minimal L‑invariant majorant: any L‑invariant kernel J≥K must dominate K∞. This minimality follows from the order‑preserving nature of L and a simple monotone convergence argument on finite Gram matrices.

A key structural result is the orthogonal realization of the defect tower. Let H_{Dₙ} be the RKHS of Dₙ and set E:=H_K⊕⊕{n≥0}H{Dₙ}. For each point s define the vector v(s):=K_s⊕⊕_{n≥0}(Dₙ)_s ∈E. Then ⟨v(s),v(t)⟩_E = K∞(s,t). The coordinate projection P₀:E→H_K yields a positive contraction A:=P₀*P₀ such that K(s,t)=⟨v(s),A v(t)⟩_E. Thus K is obtained from K∞ by a Radon‑Nikodym‑type compression, mirroring the classical RN theorem for completely positive maps on operator algebras.

The diagonal entries uₙ(s):=Kₙ(s,s) evolve under a scalar branching operator P on non‑negative functions: (Pu)(s)=∑_{i=1}^{m}u(ϕ_i(s)). Consequently uₙ = Pⁿ u₀ and the supremum h∞(s)=supₙ uₙ(s) is the minimal P‑harmonic majorant of the initial diagonal. The set X_fin={s:h∞(s)<∞} is precisely the region where K∞ remains finite on the diagonal; outside X_fin the diagonal entries diverge monotonically. The authors provide concrete Lyapunov‑type decay criteria and complementary branch‑counting conditions that allow one to decide finiteness versus blow‑up in concrete examples, linking the analysis to classical potential theory on trees and branching Markov chains.

A probabilistic reinterpretation is given via Gaussian processes. For each N, construct a centered Gaussian field G_N with covariance kernel K_N. The differences M_N:=G_N−G_{N−1} form an orthogonal martingale increment sequence; the predictable quadratic variation of the martingale (∑_{k≤N}M_k) is exactly the defect kernel D_N. When the Gram matrices of K_N stay uniformly bounded, the martingale converges in L² and its limit has covariance K∞. The compression operator A appears as the Radon‑Nikodym derivative between the Gaussian measures associated with K∞ and K.

Finally, the paper makes the implicit tree structure explicit by passing to the symbolic boundary Ω={1,…,m}^ℕ. From the minimal harmonic majorant h∞ a family of Doob‑transformed Markov measures μ_s on Ω is extracted. These measures provide a boundary representation of both the normalized iterates K_n/K_n(s,s) and the defect tower. An explicit boundary feature map embeds Ω into L²(μ_s) so that the normalized kernels become Gram kernels of boundary functions. This places the sub‑invariance problem within the broader context of path‑space measures, Martin boundaries, and Doob transforms for discrete branching systems, while staying entirely at the level of kernels and RKHS rather than operators.

In summary, the paper shows that the simple inequality LK ≥ K, when iterated over a finite branching system, forces a rich and rigid structure on the invariant completion K∞ and on the diagonal growth dynamics. The results intertwine dilation theory, tree potential theory, and Gaussian/RKHS methods, and they do so without any topological, measure‑theoretic, or spectral assumptions on X or on the maps ϕ_i. This makes the theory widely applicable to iterated function systems, non‑reversible Markov dynamics, and contractive interpolation problems where kernels appear as the primary objects.