Lifetimes and occupation times of conditioned Brownian motion

Biao Zhang, Purdue University

Abstract

We study d-dimensional Brownian motion started at a point x in a domain $\Omega$ and conditioned to first exit $\Omega$ at a fixed boundary point y of $\Omega$, which in the following we will just call conditioned Brownian motion. These processes are the fundamental h-processes, in the sense that every h-process is a mixture of them. The lifetime of a conditioned Brownian motion in $\Omega$ is the time it takes to exit $\Omega$. D. L. Burkholder (1977) found explicitly, in terms of zeros of a hyper-geometric function, the number $p(\theta$) such that the time it takes unconditioned Brownian motion, started at a point in a cone in $\IR\sp{n}$ of angle opening $\theta$, to exit from the cone, has finite $p\sp{th}$ moments if $p < p(\theta)$ and infinite $p(\theta)\sp{th}$ moment. We show that the lifetimes of conditioned Brownian motions in cones have finite $p\sp{th}$ moments if $p < \ (\theta)$+ ${d - 2}\over{2}$, and infinite $(2p(\theta)$ + ${d - 2}\over{2})\sp{th}$ moment. B. Davis (1988) gave upper bounds on the variance of the lifetime of two-dimensional Brownian motion conditioned to exit a simply connected planar domain at a given point. We extend these results to higher dimensional domains above the graphs of Lipschitz functions, and give a short and analytic proof of the existing results. Conditioned Brownian motion is intimately connected with many aspects of harmonic analysis, and in particular expectation and variance estimates of the lifetime of this conditioned motion have recently been used to study the heat kernel.

Degree

Ph.D.

Advisors

Davis, Purdue University.

Subject Area

Mathematics

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