Boundary Disintegration for Weighted Residual Energy Trees

TL;DR

This paper develops a boundary disintegration framework for weighted residual energy trees generated by positive operators and contractions, revealing how residuals evolve and can be represented via boundary measures.

Contribution

It introduces a novel boundary disintegration approach for residual energy trees, linking residual decay to boundary measures and martingale limits in a trace-class setting.

Findings

Residuals decrease to a trace-class boundary variable R_infinity

Boundary measures dominate intrinsic path measures

Explicit boundary densities relate path measures via disintegration

Abstract

We study iterated weighted residual (WR) splittings generated by a positive operator $R_{0}\in B\left(H\right)_{+}$ and a finite family of contractions $C_{1},\dots,C_{m}$ in $B\left(H\right)$. The associated residual update $R\mapsto R^{1/2}(I-C^{*}_{j}C_{j})R^{1/2}$ produces an $m$-ary energy tree of residuals $\left\{ R_{w}\right\} $ and dissipated pieces $\left\{ D_{w,j}\right\} $ indexed by finite words. From this tree we construct intrinsic path measures on the path space by biasing transitions either by a fixed quadratic form $x\mapsto\left\langle x,D_{w,j}x\right\rangle $ (defining the measures $\nu_{x}$) or, in the trace-class setting, by ${\rm tr}\left(D_{w,j}\right)$ (yielding a reference measure $\nu_{\mathrm{tr}}$). When $R_{0}\in S_{1}\left(H\right)_{+}$, we show that $\nu_{\mathrm{tr}}$ dominates the family $\left\{ \nu_{x}\right\} $ and identify $d\nu_{x}/d\nu_{\mathrm{tr}}$ as a canonical martingale limit of cylinder likelihood ratios. Along $\nu_{\mathrm{tr}}$-almost every branch the residuals decrease to a terminal trace-class random variable $R_{\infty}$, which we interpret as the WR boundary variable. We then disintegrate $\nu_{\mathrm{tr}}$ over $\sigma\left(R_{\infty}\right)$, obtaining a boundary law $\mu_{\mathrm{tr}}=\left(R_{\infty}\right)_{\#}\nu_{\mathrm{tr}}$ and conditional path measures $\left\{ \nu^{T}_{\mathrm{tr}}\right\} $. Finally, we show that each $\nu_{x}$ admits a boundary representation as a mixture of $\left\{ \nu^{T}_{\mathrm{tr}}\right\} $ with an explicit boundary density $h_{x}=d\mu_{x}/d\mu_{\mathrm{tr}}$, thereby organizing the family of intrinsic WR path measures by a single trace-biased boundary disintegration.