Subinvariant kernel dynamics
James Tian
TL;DR
This paper studies subinvariant dynamics of a positive definite kernel $K$ on a set $X$ pulled back along a finite family of maps $\phi_1,\dots,\phi_m$ via the branching operator $L$ with the inequality $L K \ge K$. Iterating yields a kernel tower $K_{n+1}=L K_n$ with defects $D_n=K_{n+1}-K_n$ under the diagonal boundedness condition $\sup_{n\ge0} K_n(s,s)<\infty$, and there is a canonical $L$-invariant completion $K_\infty$ satisfying $K_\infty \ge K$ and $L K_\infty = K_\infty$, which is minimal among all such majorants. A canonical defect decomposition is realized in an auxiliary space $\mathcal{E}$ so that $K_\infty(s,t)=\langle v(s), v(t)\rangle_{\mathcal{E}}$ and $K(s,t)=\langle v(s), A v(t)\rangle_{\mathcal{E}}$ with $A=P_0^*P_0$, giving a Radon-Nikodym compression and a multivariable dilation picture. The paper also provides probabilistic representations in terms of Gaussian defect martingales whose quadratic variation tracks the defect tower and a boundary Doob-transform construction producing boundary measures $\mu_s$ and a boundary feature map $\Psi$, yielding a boundary Gram kernel for the invariant completion. Together, the work connects RKHS, potential theory on trees, and boundary representations of branching dynamics in a measure-free setting.
Abstract
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.