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We study the averaged behavior of interfaces moving with oscillatory normal velocity that is periodic in time and stationary ergodic in space. This problem can be interpreted as a homogenization problem of a Hamilton-Jacobi equation with a positively homogeneous and time dependent Hamiltonian. In an earlier work, we studied the setting when the environment is periodic in space and random in time, and established homogenization results through the averaging of reachable sets. In the present setting, we show that the minimal travel time between two spatial points, which now depends also on a starting time, has a deterministic averaging limit that is independent of the starting time. This averaged travel time function is then shown to determine the effective behavior of the moving interfaces.

periodic and stochastic homogenization, minimal travel time, Hamilton-Jacobi equa- tions, viscosity solutions, dynamic environment, level set method, front propagation.