Mean Exit Time from Geodesic Spheres and Tubes
by
Mark Pinsky
Northwestern University, USA

Plenary Lecture

Local methods of Riemannian geometry were developed by Gray and Van Hecke to study the volume of small geodesic balls. The same ideas were used by Gray and the speaker to study the mean exit time of Brownian motion from small geodesic balls. The first basic result is a three-term asymptotic expansion of the mean exit time in powers of the radius. This is used to recognize Euclidean space from its exit time in dimension six or less. The same methods can be used to find asymptotic expansions for the harmonic measure of a small ball (Ming Liao) and the principal eigenvalue of the Dirichlet Laplacian (Karp & P). The six-dimensional counter-example of H.R. Hughes settles negatively the question of "can you feel the shape of a manifold with Brownian motion?"

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-63.