Subinvariant kernel dynamics

TL;DR

This paper investigates the behavior of positive definite kernels under self-maps, introducing a kernel tower framework with invariant majorants, and provides probabilistic and boundary representations for the defect sequences.

Contribution

It introduces a novel kernel tower structure under subinvariance, characterizes the minimal invariant majorant, and develops probabilistic and boundary models for defect sequences.

Findings

Existence of a smallest invariant majorant kernel.

Explicit construction of a canonical defect space.

Probabilistic models linking defect sequences to Gaussian martingales.

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.