  We study the averaging of fronts moving with positive oscillatory normal
velocity, which is periodic in space and stationary ergodic in time. The
problem can be reformulated as the homogenization of coercive level set
Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the
difficulties due to the oscillations in time and the sublinear growth of the
Hamiltonian, we first study the long time averaged behavior of the associated
reachable sets using geometric arguments. The results are new for higher than
one dimensions even in the space-time periodic setting.
