## Brownian motion in self-similar domains.(English)Zbl 1101.60062

Summary: For $$T\neq1$$, the domain $$G$$ is $$T$$-homogeneous if $$TG=G$$. If $$0\neq G$$, 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 $$\tau_G$$ 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 $$C^2$$.

### MSC:

 60J65 Brownian motion
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### References:

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