We consider the lifetime of a Brownian motion in a bounded planar domain. This motion starts at some point x, is conditioned to be killed at the boundary and is conditioned to converge to some point y. It has been conjectured that the expected lifetime becomes maximal for x and y both being boundary points. In this thesis, we give a counterexample and show that there are multiply connected domains, where the maximal lifetime occurs for interior points. Before we consider these domains in Part II, we need formulae describing the asymptotic behaviour of expected lifetimes on domains that consist of several subdomains connected through small gaps. In Part I, we derive limit expressions for these which are functions of expected lifetimes on the subdomains.\ud This thesis has been published in 2012 by Verlag Dr. Hut, München (ISBN 978-3-8439-0297-7)
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