Abstract: We discuss some homogenization problems of Hamilton-Jacobi equations in time-dependent (dynamic) random environments, where the  coefficients of PDEs are highly oscillatory in the space and time variables. We consider both first order and second order equations, and
 emphasize how to overcome the difficulty imposed by the lack of coercivity in the time derivative. In the first order case with linear growing Hamiltonian, periodicity in either the space or the time variable is  assumed; in the second order case with at most quadratic growing  Hamiltonian, uniform ellipticity of the second order term is assumed.