  We adapt the metric approach to the study of stationary ergodic
Hamilton-Jacobi equations, for which a notion of admissible random
(sub)solution is defined. For any level of the Hamiltonian greater than or
equal to a distinguished critical value, we define an intrinsic random
semidistance and prove that an asymptotic norm does exist. Taking as source
region a suitable class of closed random sets, we show that the Lax formula
provides admissible subsolutions. This enables us to relate the degeneracies of
the critical stable norm to the existence/nonexistence of exact or approximate
critical admissible solutions.
