Boundary Disintegration for Weighted Residual Energy Trees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917214772264960 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14646 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Boundary Disintegration for Weighted Residual Energy Trees Tian, James Probability Functional Analysis Primary: 47B65. Secondary: 28A50, 46G10, 47B10, 60G42 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. |
| title | Boundary Disintegration for Weighted Residual Energy Trees |
| topic | Probability Functional Analysis Primary: 47B65. Secondary: 28A50, 46G10, 47B10, 60G42 |
| url | https://arxiv.org/abs/2601.14646 |