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

Abstract

Bounds for the 3G-expression∫ ^Ω G(x,z)G(z,y)d,z/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 Ω. The 3G-formula equals${\mathbb{E}}_x^y (\tau _\Omega )$, the expected lifetime for a Brownian motion starting in$x \in \bar \Omega$ that is killed on exiting ω and conditioned to converge to and to be stopped at$y \in \bar \Omega$. Although it was shown by probabilistic methods for bounded (simply connected) 2d-domains that ifx ε δΩ, then the supremum ofy \at E ^y _x is assumed for somey at the boundary, the analogous question remained open forx in the interior. Here we are able to give an answer in the case thatB ⊂ ℝ is the unit disk. The dependence of this quantity on the positions ofx andy is investigated, and it is shown that indeed E ^y _x (\gt_\om) is maximized on$\bar B^2$ 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, relationE ^y _x (\gt_\om) with a ‘sum of inverse eigenvalues’ that was conjectured recently by Kawohl and Sweers.