  It was pointed out in [P.L. Lions, G. Papanicolaou, S. Varadhan,
Homogenization of Hamilton-Jacobi equation, unpublished preprint (1987)] that,
for first order Hamilton-Jacobi (HJ) equations, homogenization starting with
affine initial data implies homogenization for general uniformly continuous
initial data. The argument makes use of some properties of the HJ semi-group,
in particular, the finite speed of propagation. The last property is lost for
viscous HJ equations. In this paper we prove the above mentioned implication in
both viscous and non-viscous cases. Our proof relies on a variant of Evans's
perturbed test function method. As an application, we show homogenization in
the stationary ergodic setting for viscous and non-viscous HJ equations in one
space dimension with non-convex Hamiltonians of specific form. The results are
new in the viscous case.
