02-32
On a conditioned Brownian motion and a maximum principle in the disk
by A. Dall\'Acqua; H.-Ch. Grunau; G.H. Sweers
Preprint series: 02-32, Preprints
The paper is published: J. Anal. 93 , 309-329 (2004).
Abstract: We study the expected lifetime $\mathbb{E}_{x}^{y}\left( \tau _{B}\right) $ for a Brownian motion starting in $x \in \bar{B},$ conditioned to converge to and stopped at $y\in \bar{B}$ that is killed on exiting $B$. Here $B \subset \mathbb{R}^2$ is the unit disk. The dependence of this quantity on the positions $x$ and $y$ is investigated and it is proved that indeed $\mathbb{E}_{x}^{y}\left( \tau _{B}\right)$ is maximized on $\bar{B}^{2}$ by opposite boundary points. In turn this gives an answer to a number of questions related with the best constant for the positivity preserving property of some elliptic systems. The relation with such systems comes through a so-called 3G-expression.
Keywords: Conditioned Brownian motion, maximum of the expected lifetime, maximum principles, 3-G-quotient, Moebius transforms
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