Brownian exit-time expectation in an orthant
= Brownian exit-time expectation in an orthant
{c}
{title2=$\mathbb E_xT_{(0,\infty)^n}<\infty\iff n>2$}
For <independent> coordinates of standard <Brownian motion> and $x_i>0$, the <Brownian reflection principle> gives $\mathbb P_x(T>t)=\prod_i[2\Phi(x_i/\sqrt t)-1]$. This is asymptotic to $(2/\pi)^{n/2}\prod_ix_i\,t^{-n/2}$. The <tail integral formula for expectation> therefore proves the stated criterion, with logarithmic divergence in dimension two.