Brownian motion in self-similar domains

Abstract

For $T\ne 1$ , the domain G is T-homogeneous if TG=G. If $0\neg \in G$ , then necessarily $0\in \partial G$ . It is known that for some p>0, the Martin kernel K at infinity satisfies $K\left(\mathrm{Tx}\right)=T^{\mathrm{pK}}\left(x\right)$ for all $x\in G$ . We show that in dimension $d\ge 2$ , if G is also Lipschitz, then the exit time τ_G of Brownian motion from G satisfies $P_x\left({\tau }_G>t\right)\approx K\left(x\right)t^{-p/2}$ as $t\to \mathrm{\infty }$ . 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.