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

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