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2002-32

On a conditioned Brownian motion and a maximum principle in the disk

by A. Dall'Acqua, H.-Ch. Grunau, G.H. Sweers.


Series: 2002-32, Preprints

MSC:
35B50 Maximum principles
60J65 Brownian motion
35A08 Fundamental solutions

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

This paper was published in:
J. Anal. 93 , 309-329 (2004).