On the lifetime of a conditioned Brownian motion in the ball

Consider the Brownian motion conditioned to start in x, to converge to y, with x,y ∈ Ω, and to be killed at the boundary ∂Ω. Here Ω is a bounded domain in Rn. Forwhich x and y is the lifetime of this Brownian motion maximal? One would guess for x and y being opposite boundary points and we will show that this holds true for balls in Rn. As a consequence we find the best constant for the positivity preserving property of some elliptic systems and an identity between this constant and a sum of inverse Dirichlet eigenvalues. © 2007 Elsevier Inc. All rights reserved