Boundary Disintegration for Weighted Residual Energy Trees
James Tian
TL;DR
This work develops a probabilistic operator-driven framework for iterated weighted residual (WR) splittings generated by a positive operator $R_0$ and contractions $\{C_j\}$, forming an energy tree of residuals $\{R_w\}$ and dissipations $\{D_{w,j}\}$ and attaching two intrinsic path measures $\nu_x$ and $\nu_{tr}$ on absorbed paths. Under the trace-class hypothesis $R_0\in S_1(H)_+$, $\nu_{tr}$ dominates $\nu_x$ and $d\nu_x/d\nu_{tr}$ is the canonical $\nu_{tr}$-martingale limit; the boundary variable $R_\infty$ yields a disintegration $\nu_{tr}=\int \nu^{T}_{tr} \, d\mu_{tr}(T)$ and a boundary representation $\nu_x = \int h_x(T) \nu^{T}_{tr} \, d\mu_{tr}(T)$. The paper proves extinction estimates for both bias schemes, constructs scalar and operator-valued Radon-Nikodym densities, and presents a boundary disintegration that organizes the family of intrinsic WR path measures into a unified trace-biased description. Overall, the WR boundary framework clarifies the asymptotic behavior of residuals and provides a canonical change-of-measure principle on WR path space, with explicit connections between edge energies, trace dissipation, and boundary data.
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 $ν_{x}$) or, in the trace-class setting, by ${\rm tr}\left(D_{w,j}\right)$ (yielding a reference measure $ν_{\mathrm{tr}}$). When $R_{0}\in S_{1}\left(H\right)_{+}$, we show that $ν_{\mathrm{tr}}$ dominates the family $\left\{ ν_{x}\right\} $ and identify $dν_{x}/dν_{\mathrm{tr}}$ as a canonical martingale limit of cylinder likelihood ratios. Along $ν_{\mathrm{tr}}$-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 $ν_{\mathrm{tr}}$ over $σ\left(R_{\infty}\right)$, obtaining a boundary law $μ_{\mathrm{tr}}=\left(R_{\infty}\right)_{\#}ν_{\mathrm{tr}}$ and conditional path measures $\left\{ ν^{T}_{\mathrm{tr}}\right\} $. Finally, we show that each $ν_{x}$ admits a boundary representation as a mixture of $\left\{ ν^{T}_{\mathrm{tr}}\right\} $ with an explicit boundary density $h_{x}=dμ_{x}/dμ_{\mathrm{tr}}$, thereby organizing the family of intrinsic WR path measures by a single trace-biased boundary disintegration.