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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 3 open journal systems 


Exit Time, Green Function and Semilinear Elliptic Equations

Rami Atar, Technion
Siva Athreya, Indian Statistical Institute
Zhen-Qing Chen, University of Washington


Abstract
Let $D$ be a bounded Lipschitz domain in $R^n$ with $ngeq 2$ and $tau_D$ be the first exit time from $D$ by Brownian motion on $R^n$. In the first part of this paper, we are concerned with sharp estimates on the expected exit time $E_x [ tau_D]$. We show that if $D$ satisfies a uniform interior cone condition with angle $theta in ( cos^{-1}(1/sqrt{n}), , pi )$, then $c_1 varphi_1(x) leq E_x [tau_D] leq c_2 varphi_1 (x)$ on $D$. Here $varphi_1$ is the first positive eigenfunction for the Dirichlet Laplacian on $D$. The above result is sharp as we show that if $D$ is a truncated circular cone with angle $theta < cos^{-1}(1/sqrt{n})$, then the upper bound for $E_x [tau_D]$ fails. These results are then used in the second part of this paper to investigate whether positive solutions of the semilinear equation $Delta u = u^{p}$ in $ D,$ $pin R$, that vanish on an open subset $Gamma subset p D$ decay at the same rate as $varphi_1$ on $Gamma$.


Full text: PDF

Pages: 50-71

Published on: January 14, 2009
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Electronic Journal of Probability. ISSN: 1083-6489