  We present qualitative and quantitative homogenization results for pathwise
Hamilton-Jacobi equations with "rough" multiplicative driving signals. When
there is only one such signal and the Hamiltonian is convex, we show that the
equation, as well as equations with smooth approximating paths, homogenize. In
the multi-signal setting, we demonstrate that blow-up or homogenization may
take place. The paper also includes a new well-posedness result, which gives
explicit estimates for the continuity of the solution map and the
equicontinuity of solutions in the spatial variable.
