Infinite-Dimensional Levy Area and Spectral Ancestry

Abstract

We relate primitive ancestry in free-Lie expansions of infinite-dimensional Lévy area to critical weak-Schatten spectra. For $L=\mathfrak{L}(V)$, $D=[L,L]$, and $J=\ker (T(V)\to S(V))$, we prove $L\cap J^r={\gamma }_r(D)$ and identify the exact normal layers with ${\mathfrak{L}}_r(D/[D,D])$. Continuous Hilbert-martingale area has the sharp split endpoint $H\times {\mathcal{S}}_{1,\infty }^0(H{)}_{\mathrm{skew}}\times {\mathcal{S}}_1(H{)}_{\mathrm{sa}}$. The critical grade $r$ has singular profile $(\log (e+N){)}^{r-1}/N$. A Rees response kernel characterizes strictness, equality of ancestry and spectral depth, and recovery of the finite labelled ancestry flag. Pure-area Hall experiments give full-support kernels and exact critical laws. Revealed brackets and evolution families provide conditional covariance and second-order Lévy/SPDE interfaces.

66 pages