  We consider the specified stochastic homogenization of first order evolutive
Hamilton-Jacobi equations on a very simple junction, i.e the real line with a
junction at the origin. Far from the origin, we assume that the considered
hamiltonian is closed to given stationary ergodic hamiltonians (which are
different on the left and on the right). Near the origin, there is a
perturbation zone which allows to pass from one hamiltonian to the other. The
main result of this paper is a stochastic homogenization as the length of the
transition zone goes to zero. More precisely, at the limit we get two
deterministic right and left hamiltonians with a deterministic junction
condition at the origin. The main difficulty and novelty of the paper come from
the fact that the hamiltonian is not stationary ergodic. Up to our knowledge,
this is the first specified stochastic homogenization result. This work is
motivated by traffic flow applications.
