Geometric & Functional Analysis GAFA

, Volume 11, Issue 4, pp 651–692

Comparison theorems for exit times

Authors

  • A. Burchard
    • Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA, e-mail: burchard@virginia.edu
  • M. Schmuckenschläger
    • Institut für Analysis und Numerik, Johannes Kepler Universität Linz, A-4040 Linz, Austria, e-mail: michael.schmuckenschlaeger@telering.at

DOI: 10.1007/PL00001681

Cite this article as:
Burchard, A. & Schmuckenschläger, M. GAFA, Geom. funct. anal. (2001) 11: 651. doi:10.1007/PL00001681

Abstract.

We study bounds on the exit time of Brownian motion from a set in terms of its size and shape, and the relation of such bounds with isoperimetric inequalities. The first result is an upper bound for the distribution function of the exit time from a subset of a sphere or hyperbolic space of constant curvature in terms of the exit time from a disc of the same volume. This amounts to a rearrangement inequality for the Dirichlet heat kernel. To connect this inequality with the classical isoperimetric inequality, we derive a formula for the perimeter of a set in terms of the heat flow over the boundary. An auxiliary result generalizes Riesz' rearrangement inequality to multiple integrals.

Copyright information

© Birkhäuser Verlag, Basel 2001