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

Comparison of exit moment spectra for extrinsic metric balls