Infinite-Dimensional Levy Area: Sharp Operator-Ideal Regularity and Anisotropic Ito Geometry

Abstract

We prove a sharp universal operator-ideal theorem for infinite-dimensional L\'evy area. Every fixed area increment of a continuous locally square-integrable Hilbert-space martingale belongs almost surely to the separable little weak trace class \(\mathcal S^0_{1,\infty}\). Deterministic trace clocks yield dimension-free weak-trace estimates, and deterministic terminal trace bounds yield finite-rank approximation. Gaussian trace-clock examples establish Lorentz--Schatten optimality and arbitrarily slow little-weak singular-value decay. We further identify three second-order regularity mechanisms. Covariance spectra determine finer model-dependent fixed-time ideals; temporal smoothing and a quasi-Banach Chen--Kolmogorov principle yield endpoint path closure and Galerkin convergence; and, in the flat identity sector, trace-sensitive observables select an anisotropic split between skew area and trace-class symmetric compensation, with a split-dual integral consistent with geometric and classical It\^o integration.