@article{EJP597,
author = {Rami Atar and Siva Athreya and Zhen-Qing Chen},
title = {Exit Time, Green Function and Semilinear Elliptic Equations},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {14},
year = {2009},
keywords = {Brownian motion; exit time; Feynman-Kac transform; Lipschitz domain; Dirichlet Laplacian; ground state; boundary Harnack principle; Green function estimates; semilinear elliptic equation; Schauder's fixed point theorem},
abstract = {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 $\Gamma$.},
pages = {no. 3, 50-71},
issn = {1083-6489},
doi = {10.1214/EJP.v14-597},
url = {http://ejp.ejpecp.org/article/view/597}}