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 $u_0$.