The Annals of Probability

The Asymptotic Behavior of the Solution of the Exterior Dirichlet Problem for Brownian Motion Perturbed by a Small Parameter Drift

Ross G. Pinsky

Full-text: Open access

Abstract

Let $L_\varepsilon = \frac{1}{2}\Delta + \varepsilon b \cdot \nabla$ in $R^d, d \geq 3$, generate a recurrent diffusion for each $\varepsilon > 0$, where $b \in C^\alpha(R^d)$, and let $D \subset R^d$ be an exterior domain. Then by the recurrence assumption, for each $\psi \in C(\partial D)$, there exists a unique solution in the class of bounded solutions to the Dirichlet problem $L_\varepsilon u_\varepsilon = 0$ in $D$ and $u_\varepsilon = \psi$ on $\partial D$. On the other hand, by the transience of $d$-dimensional Brownian motion, there is no uniqueness in the class of bounded solutions for the Dirichlet problem $\frac{1}{2} \Delta u = 0$ in $D$ and $u = \psi$ on $\partial D$. Since the Martin boundary at $\infty$ for Brownian motion consists of a single point, uniqueness is obtained by adding the condition $\lim_{|x|\rightarrow\infty} u(x) = c$. We show that $u_0(x) \equiv \lim_{\varepsilon\rightarrow 0} u_\varepsilon(x)$ exists and satisfies $\frac{1}{2}\Delta u_0 = 0$ in $D, u_0 = \psi$ on $\partial D$ and $\lim_{|x|\rightarrow\infty} u_0(x) = c$, where $c$ is given as follows. Let $P^h_x$ denote the measure associated with Doob's conditioned Brownian motion conditioned to exit $D$ at $\partial D$ rather than at $\infty$. Let $\tau = \inf\{t \geq 0: X(t) \in \partial D\}$ and define the harmonic measure $u^h_x(dy) = P^h_x(X(\tau) \in dy)$. Then $\mu^h_\infty \equiv \lim_{|x|\rightarrow\infty} \mu^h_x$ exists and $c = \int_{\partial D}\psi(y)\mu^h_\infty(dy)$. We also show that the energy integral $\int_D|\nabla u|^2 dx$, when varied over all bounded functions $u \in W^{1,2}_{\operatorname{loc}}(D)$ which satisfy $u = \psi$ on $\partial D$, takes on its minimum uniquely at $u_0$.

Article information

Source
Ann. Probab., Volume 18, Number 4 (1990), 1602-1618.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990635

Digital Object Identifier
doi:10.1214/aop/1176990635

Mathematical Reviews number (MathSciNet)
MR1071812

Zentralblatt MATH identifier
0715.60095

JSTOR
links.jstor.org

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 35B20: Perturbations

Keywords
Exterior Dirichlet problem Brownian motion harmonic measure small parameter drift

Citation

Pinsky, Ross G. The Asymptotic Behavior of the Solution of the Exterior Dirichlet Problem for Brownian Motion Perturbed by a Small Parameter Drift. Ann. Probab. 18 (1990), no. 4, 1602--1618. doi:10.1214/aop/1176990635. https://projecteuclid.org/euclid.aop/1176990635


Export citation