For $T\neq 1$ , the domain G is T-homogeneous if TG=G. If $0\not\in G$ , then necessarily $0\in\partial G$ . It is known that for some p>0, the Martin kernel K at infinity satisfies $K(Tx)=T^pK(x)$ for all $x\in G$ . We show that in dimension $d\geq 2$ , if G is also Lipschitz, then the exit time τG of Brownian motion from G satisfies $P_x(\tau_G>t) \approx K(x)t^{-p/2}$ as $t\to\infty$ . An analogous result holds for conditioned Brownian motion, but this time the decay power is $1-p-d/2$ . In two dimensions, we can relax the Lipschitz condition at 0 at the expense of making the rest of the boundary C2.