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

Abstract

Bounds for the 3G-expression∫ΩG(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 e δΩ, then the supremum ofy \at Exy 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 Exy(\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, relationExy(\gt\om) with a ‘sum of inverse eigenvalues’ that was conjectured recently by Kawohl and Sweers.