The asymptotic behavior of the solution of the exterior Dirichlet problem for Brownian motion perturbed by a small parameter drift. (English) Zbl 0715.60095

Pinsky, Ross G.

Ann. Probab. 18, No. 4, 1602-1618 (1990).

Consider the family of diffusions in \({\mathbb{R}}^ d\), \(d\geq 3\), which are generated by \(L_{\epsilon}=(1/2)\Delta +\epsilon b\nabla\), \(\epsilon >0\), where \(b\in {\mathcal C}^{\infty}({\mathbb{R}}^ d)\), and assume that for every \(\epsilon\), the diffusion is recurrent. Let \(D\subset {\mathbb{R}}^ d\) be an exterior domain (i.e. the complement of a compact set) with Lipschitz boundary, and \(\Psi\) a continuous function on \(\partial D\). By the recurrence assumption, for each \(\epsilon >0\), there is a unique bounded solution to the Dirichlet problem \(L_{\epsilon}u=0\) in D, \(u=\Psi\) on \(\partial D.\)
The author shows that \(\lim_{\epsilon \to 0}u_{\epsilon}(x)=u_ 0(x)\) exists. The function \(u_ 0\) is the unique solution to \(\Delta u=0\) in D, \(u=\Psi\) on \(\partial D\) and \(\lim_{| x| \to \infty}u(x)=c\) (this last condition stems from the fact that the Martin boundary at \(\infty\) for a d-dimensional Brownian motion consists of a single point), where \(c=\int_{\partial D}\Psi (y)\mu^ h_{\infty}(dy)\), \(\mu^ h_{\infty}\) being the weak limit as \(| x|\) goes to infinity of the harmonic measures \(\mu^ h_ x\) associated with Brownian motion in D conditioned (in the sense of Doob) on exiting D at \(\partial D\) rather than at \(\infty\). The function \(u_ 0\) also appears in a variational problem: Let \(J(u)=\int | \nabla u|^ 2du\) be the energy integral of u. Then \(u_ 0\) is the unique bounded function in \(W^{1,2}(D)\), with \(u=\Psi\) on \(\partial D\) at which J(\(\cdot)\) attains its minimum.

Reviewer: J.Bertoin

MSC:

60J60	Diffusion processes
35B20	Perturbations in context of PDEs

Keywords:

exterior domain; Lipschitz boundary; recurrence assumption; Dirichlet problem; Martin boundary; energy integral