# Homogenization of Hamilton-Jacobi equations in dynamic random environments

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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.