Homogenization of pathwise Hamilton-Jacobi equations

We consider homogenization problems for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. The main result is for a single path and with a Hamiltonian that is smooth, convex, and positively homogenous. For such equations, we discuss some recent methods for obtaining well-posedness and stability estimates. We also address equations involving multiple driving signals, and show that homogenization or blow-up can occur.