### Perturbation theory for random walk in asymmetric random environment.

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In this paper we continue the study of the Dirichlet problem for an elliptic equation on a domain in R^{3} which was begun in [5]. For R > 0 let Ω_{R} be the ball of radius R centered at the origin with boundary ∂Ω _{R}. The Dirichlet problem we are concerned with is the following:
(-Δ - b(x).∇) u(x) = f(x), x ∈ Ω _{R},
with zero boundary conditions
...

In this paper we are concerned with studying the Dirichlet problem for an elliptic equation on a domain in R^{3}. For simplicity we shall assume that the domain is a ball Ω_{R} of radius R. Thus:
Ω_{R} = {x ∈ R^{3} : |x| < R}.
The equation we are concerned with is given by
(-Δ - b(x).∇) u(x) = f(x), x ∈ Ω_{R},
with zero Dirichlet boundary conditions.

Consider 3-dimensional Brownian motion started on the unit sphere {|x| = 1} with initial density ρ. Let ρt be the first hitting density on the sphere {|x| = t + 1}, t > 0. Then the linear operators T defined by T ρ = ρ form a semigroup with an infinitesimal generator which is approximately the square root of the Laplacian. This paper studies the analogous situation for Brownian motion with a drift , where is small in a suitable scale invariant norm.

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