Gaussian Exterior Dilation and Schatten Classes of Lévy Area
Abstract
We study the skew-adjoint Hilbert--Schmidt operator obtained by evaluating a fixed Hilbert two-form on a centered Gaussian linear map whose covariance is the inner product on its domain multiplied by a fixed nonnegative trace-class operator. We compute the expected regularized determinant of the resulting random operator and compare the random determinant with this expectation by dilation of the scalar parameter. For each fixed covariance and two-form, a single dilation almost surely applies to every nonnegative parameter value and yields simultaneous bounds for all Schatten norms and quasi-norms of orders 0<p<2. The least admissible dilation satisfies a universal weak-L1 estimate with optimal tail exponent. For Wiener area, a rank-one perturbation identity identifies the mean determinant with the regularized determinant of an explicit deterministic skew-adjoint operator and reduces Schatten membership to conditions on the covariance eigenvalues. Consequently, for each 0<p<1, the area belongs almost surely to the Schatten class of order p if and only if the covariance of the Wiener process does. At p=1, the criterion is finite covariance entropy; for p>1, trace-class covariance suffices.
Related
- Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes II
- Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols
- γ-Valued Stein--Tomas Restriction for Schatten Classes
- On Surjective Max-Phase Isometries of Schatten p-Classes (0<p<1)
- Stable and Gaussian Fluctuation Limit of a Lévy-Driven Slow-Fast System with Polynomial Dissipation
BibTeX
@misc{zhao2026gaussian,
title = {{Gaussian Exterior Dilation and Schatten Classes of Lévy Area}},
author = {Guangqian Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34984},
eprint = {2610.34984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34984},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian Exterior Dilation and Schatten Classes of Lévy Area
AU - Guangqian Zhao
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34984
UR - https://arxiv.org/abs/2610.34984
AB - We study the skew-adjoint Hilbert--Schmidt operator obtained by evaluating a fixed Hilbert two-form on a centered Gaussian linear map whose covariance is the inner product on its domain multiplied by a fixed nonnegative trace-class operator. We compute the expected regularized determinant of the resulting random operator and compare the random determinant with this expectation by dilation of the scalar parameter. For each fixed covariance and two-form, a single dilation almost surely applies to every nonnegative parameter value and yields simultaneous bounds for all Schatten norms and quasi-norms of orders 0<p<2. The least admissible dilation satisfies a universal weak-L^1 estimate with optimal tail exponent. For Wiener area, a rank-one perturbation identity identifies the mean determinant with the regularized determinant of an explicit deterministic skew-adjoint operator and reduces Schatten membership to conditions on the covariance eigenvalues. Consequently, for each 0<p<1, the area belongs almost surely to the Schatten class of order p if and only if the covariance of the Wiener process does. At p=1, the criterion is finite covariance entropy; for p>1, trace-class covariance suffices.
ER -