On the lifetime of conditioned Brownian motion
Abstract
This dissertation studies the Brownian motion started at a point x in a domain D in $R\sp d$ and conditioned to exit D at a point y of its boundary $\partial D$. Especially the lifetime T(x, y) of the motion is studied. In 1983, M. Cranston and T. McConnell proved a conjecture of Chung by showing that there is an absolute constant c such that$$\sup\sb{x\in D,y\in\partial D}\ E\ T(x, y) < c\ Area(D),\leqno{(1)}$$if D is a domain of the plane. The thesis studies the converse in equality and shows there is no constant c such that$$\sup\sb{x\in D,y\in\partial D}\ E\ T(x, y) > c\ Area(D),\leqno{(2)}$$by giving an example of a domain of infinite area in which the expected lifetime of any conditioned Brownian motion is less than 1. It is also shown that (2) holds, for a universal constant c, for all convex domains D in the plane.
Degree
Ph.D.
Subject Area
Mathematics
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