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Exit Time, Green Function and Semilinear Elliptic Equations

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Exit Time, Green Function and Semilinear Elliptic Equations
2.	Creator	Author's name, affiliation, country	Rami Atar; Technion
2.	Creator	Author's name, affiliation, country	Siva Athreya; Indian Statistical Institute
2.	Creator	Author's name, affiliation, country	Zhen-Qing Chen; University of Washington
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	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
3.	Subject	Subject classification	60H30; 60J45; 35J65; 60J35; 35J10
4.	Description	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 $\Gamma$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF; CSIR
7.	Date	(YYYY-MM-DD)	2009-01-14
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/597
10.	Identifier	Digital Object Identifier	10.1214/EJP.v14-597
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 14
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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