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Homogenization of diffusion processes in tubular domains

We consider a tubular domain D ⊂ R about the x-axis with variable cross-section: D = {(x, y): x ∈ R, ε−1y ∈ Dx}, where 0 < ε ≪ 1, and the cross-section Dx is a bounded domain in R containing the origin. It is assumed that the domain D = D ∣∣ ε=1 has a sufficiently smooth boundary ∂D and that the normal vector to ∂D is never parallel to the x-axis. Let Z t = (X ε t , Y ε t ) denote a Wiener process in D ε with normal reflection on the boundary. The sample paths of this process can be described by the equations