  This paper analyzes two classes of second order level set PDE in periodic
media in the parabolic scaling. First, we study fully nonlinear geometric
operators under general assumptions in dimension $d = 2$ and prove that the
associated equations homogenize in this case. Next, we treat a class of
quasi-linear geometric operators in arbitrary dimensions $d \geq 2$. In this
setting, by adapting arguments form the study of oscillating boundary value
problems, we prove that the effective coefficients are generically
discontinuous in all dimensions $d \geq 3$. This necessitates a study of level
set PDE driven by operators that are discontinuous at every rational direction
on the sphere. We prove that, in fact, the effective operators so obtained do
have a comparison principle and, thus, homogenization occurs. Finally, we
investigate the connection between the effective mobility obtained in the
quasi-linear case and linear response, drawing a connection between our results
and those obtained in the hyperbolic scaling.
