Breakdown of homogenization for the random Hamilton-Jacobi equations

We study the homogenization of Lagrangian functionals of Hamilton-Jacobi equations (HJ) with quadratic nonlinearity and unbounded stationary ergodic random potential in R^d, d ≥ 1. We show that homogenization holds if and only if the potential is bounded from above. When the potential is unbounded from above, homogenization breaks down, due to the almost sure growth of the running maxima of the random potential. If the unbounded randomness appears in the advection term, homogenization may or may not hold depending on the structure of the flow field. In (compressible) unbounded gradient flows, homogenization holds in spite of the unboundedness. In (incompressible) unbounded shear flows, homogenization breaks down again due to unbounded running maxima of the flows. Results for random advection follow from a transformation of the problem to that of HJ with random potential. Analogous effective behavior is present for front speeds in reaction-diffusion-advection equations with unbounded random advection, and may have broader implications for wave propagation in random media.