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

Bounds for the 3G-expression integral(Omega) G (x, z) G (z, y) dz/G (x, y) play a fundamental role in potential theory. Here, G (x, y) is the Green function for the Laplace problem with zero Dirichlet boundary conditions on Q. The 3G-formula equals E-x(y) (tau(Omega)), the expected lifetime for a Brownian motion starting in x is an element of (Ω) over bar that is killed on exiting Q and conditioned to converge to and to be stopped at y is an element of (Ω) over bar. Although it was shown by probabilistic methods for bounded (simply connected) 2d-domains that if x is an element of deltaOmega, then the supremum of y --> E-x(y) (tau(Omega)) is assumed for some y at the boundary, the analogous question remained open for x in the interior. Here we are able to give an answer in the case that B subset of R-2 is the unit disk. The dependence of this quantity on the positions of x and y is investigated, and it is shown that indeed E-x(y) (tau(B)) is maximized on (B) over bar (2) by opposite boundary points. The result also gives an answer to a number of questions related to the best constant for the positivity-preserving property of some elliptic systems. In particular, it confirms a relation of E-x(y) (tau(B)) with a 'sum of inverse eigenvalues' that was conjectured recently by Kawohl and Sweers.