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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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