  This paper deals with homogenization of second order divergence form
parabolic operators with locally stationary coefficients. Roughly speaking,
locally stationary coefficients have two evolution scales: both an almost
constant microscopic one and a smoothly varying macroscopic one. The
homogenization procedure aims to give a macroscopic approximation that takes
into account the microscopic heterogeneities. This paper follows "Diffusion in
a locally stationary random environment" (published in Probability Theory and
Related Fields) and improves this latter work by considering possibly
degenerate diffusion matrices. The geometry of the homogenized equation shows
that the particle is trapped in subspace of R^d.
