Infinite-Dimensional Levy Area and Spectral Ancestry

Abstract

We relate primitive ancestry in free-Lie expansions of infinite-dimensional L\'evy 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 $H\times\mathcal S^0_{1,\infty}(H)_{\rm skew}\times\mathcal S_1(H)_{\rm sa}$. 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 L\'evy/SPDE interfaces.