## The Annals of Probability

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

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

#### 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