Comparison of exit moment spectra for extrinsic metric balls

TL;DR

This paper establishes explicit bounds for the exit time spectra of Brownian motion from extrinsic metric balls in submanifolds, using curvature controls and comparison with model spaces, leading to new intrinsic spectral results.

Contribution

It provides sharp upper and lower bounds for exit moment spectra in submanifolds with controlled curvature, extending spectral comparison to intrinsic and extrinsic settings.

Findings

Bounds are sharp in characteristic cases.

New intrinsic comparison results for ambient manifold balls.

Explicit bounds relate spectra to model space geodesic balls.

Abstract

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in $N$. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds $N^n$ themselves.