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Mean exit time from convex hypersurfaces

Author: Vicente Palmer
Journal: Proc. Amer. Math. Soc. 126 (1998), 2089-2094
MSC (1991): Primary 53C21, 58G32
MathSciNet review: 1443163
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Abstract: L. Karp and M. Pinsky proved that, for small radius $R$, the mean exit time function $E_{R}$ of an extrinsic $R$-ball in a hypersurface $P^{n-1} \subseteq \mathbb{R}^{n}$ is bounded from below by the corresponding function $\widetilde E_{R}$ defined on an extrinsic $R$-ball in $\mathbb{R}^{n-1}$. A counterexample given by C. Mueller proves that this inequality doesn't holds in the large. In this paper we show that, if $P$ is convex, then the inequality holds for all radii. Moreover, we characterize the equality and show that analogous results are true in the sphere.

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Additional Information

Vicente Palmer
Affiliation: Departament de Matematiques, Universitat Jaume I, Castello, Spain

Keywords: Brownian motion, mean exit time, convex hypersurface, extrinsic ball
Received by editor(s): August 6, 1996
Received by editor(s) in revised form: December 10, 1996
Additional Notes: Work partially supported by a DGICYT Grant No. PB94-0972
Communicated by: Peter Li
Article copyright: © Copyright 1998 American Mathematical Society