Maxima for the Expectation of the Lifetime of a Brownian Motion in the Ball

Abstract

Let G _B (x, y) be the Green’s function of the unit ball B in ${\mathbb{R}^n, n \ge 3,}$ and ${\Gamma_B (x,y)=\int_BG_B(x, z)G_B(z, y)dz}$ the iterated Green’s function. The function

$$E_x^y(\tau_B) = \frac{\Gamma_B(x, y)}{G_B(x, y)}$$

is the expectation of the lifetime of a Brownian motion starting at ${x \in \overline{B}}$, killed on exiting B and conditioned to converge to and to be stopped at ${y \in \overline{B}}$. The aim of the paper is to prove that

$$\sup_{x \in \partial B,y \in B} E_x^y(\tau_B) = \sup_{x,y \in \partial B} E_x^y(\tau_B) = E_{x_0}^{-x_0}(\tau_B), x_0 \in\partial B$$

and that the maximum value of ${E_x^y(\tau_B)}$ occurs if and only if x, y are diametrically opposite points on the boundary of B.