Electronic Journal of Probability

Exit Time, Green Function and Semilinear Elliptic Equations

Rami Atar, Siva Athreya, and Zhen-Qing Chen

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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

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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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