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

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
Full Text: