Boundary-layer asymptotics for Gaussian-smoothed singular measures

2026-07-05

AI Summary

This work investigates the asymptotic behavior of heat kernel regularization of probability measures on manifolds with corners under Gaussian smoothing in the small-noise limit, with particular emphasis on the geometric structure near boundaries and corner points. By introducing conical boundary layers, scale renormalization, and linearization via tangent cones, the local support set is approximated by its inner tangent cone, within which a second-order expansion of the heat-regularized density is derived. The study systematically reveals, for the first time, how low-dimensional support, boundary faces, corners, and ambient curvature are encoded in the singular differential structure of the regularized score function, establishing a localized asymptotic theory. The results include uniform asymptotic expressions for the score function, the log-Hessian, and their scale derivatives, with nonlocal effects shown to decay exponentially.

Abstract

We study the small-noise asymptotics of Euclidean heat regularizations of probability measures supported on manifolds with corners. Near a boundary or corner stratum, the relevant regime is a conical boundary layer in which the observation point approaches the stratum at the same scale as the Gaussian smoothing parameter. After rescaling this layer, the support is replaced to leading order by its inward tangent cone. We prove a two-term expansion for the heat-regularized density in this regime. The leading coefficient is the Gaussian mass of the linearized cone, weighted by the density on the support and by the adapted corner Jacobian; the first correction records the variation of the density, the Jacobian, and the quadratic geometry of the embedding. A localization argument then yields the corresponding expansion for the full heat regularization, with the nonlocal contribution exponentially small. From this density expansion we derive logarithmic asymptotics and uniform expansions for the score, the log-Hessian, and the scale derivative of the score. These formulas show how lower-dimensional support, boundary faces, corners, and curvature are encoded in the singular differential structure of small-noise Gaussian regularizations.

Problem

• boundary-layer asymptotics
• Gaussian smoothing
• singular measures
• manifolds with corners
• heat regularization

Innovation

• boundary-layer asymptotics
• Gaussian smoothing
• manifolds with corners
• heat regularization
• tangent cone