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).