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 $Gamma$.
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