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Brownian Motion in Self-Similar Domains

Dante Deblassie and Robert Smits
Bernoulli
Vol. 12, No. 1 (Feb., 2006), pp. 113-132
Stable URL: http://www.jstor.org/stable/25464791
Page Count: 20
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Brownian Motion in Self-Similar Domains
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Abstract

For T ≠ 1, the domain G is T-homogeneous if TG = G. If 0 ∉ G, then necessarily 0 ∈ ∂G. It is known that for some p > 0, the Martin kernel K at infinity satisfies $K(Tx)=T^{p}K(x)$ for all x ∈ G. We show that in dimension d ≥ 2, if G is also Lipschitz, then the exit time $\tau _{G}$ of Brownian motion from G satisfies $P_{x}(\tau _{G}>1)\approx K(x)t^{-p/2}$ as t → ∞. 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.

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