Electronic Journal of Probability

Exit Time, Green Function and Semilinear Elliptic Equations

Rami Atar, Siva Athreya, and Zhen-Qing Chen

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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 $\Gamma$.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 3, 50-71.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819464

Digital Object Identifier
doi:10.1214/EJP.v14-597

Mathematical Reviews number (MathSciNet)
MR2471659

Zentralblatt MATH identifier
1190.60056

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 35J65: Nonlinear boundary value problems for linear elliptic equations 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 35J10: Schrödinger operator [See also 35Pxx]

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Atar, Rami; Athreya, Siva; Chen, Zhen-Qing. Exit Time, Green Function and Semilinear Elliptic Equations. Electron. J. Probab. 14 (2009), paper no. 3, 50--71. doi:10.1214/EJP.v14-597. https://projecteuclid.org/euclid.ejp/1464819464


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