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2009 Exit Time, Green Function and Semilinear Elliptic Equations
Rami Atar, Siva Athreya, Zhen-Qing Chen
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Electron. J. Probab. 14: 50-71 (2009). DOI: 10.1214/EJP.v14-597


Let $D$ be a bounded Lipschitz domain in $R^n$ with $n\geq 2$ 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,$ $p\in R$, that vanish on an open subset $\Gamma \subset \partial D$ decay at the same rate as $\varphi_1$ on $\Gamma$.


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Rami Atar. Siva Athreya. Zhen-Qing Chen. "Exit Time, Green Function and Semilinear Elliptic Equations." Electron. J. Probab. 14 50 - 71, 2009.


Accepted: 14 January 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1190.60056
MathSciNet: MR2471659
Digital Object Identifier: 10.1214/EJP.v14-597

Primary: 60H30
Secondary: 35J10 , 35J65 , 60J35 , 60J45

Keywords: boundary Harnack principle , Brownian motion , Dirichlet Laplacian , Exit time , Feynman-Kac transform , Green function estimates , ground state , Lipschitz domain , Schauder's fixed point theorem , semilinear elliptic equation

Vol.14 • 2009
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