Stochastic homogenization of interfaces moving with changing sign velocity

We are interested in the averaging behavior of interfaces moving in stationary ergodic environments with oscillatory normal velocity which changes sign. The problem can be reformulated as the homogenization of a Hamilton–Jacobi equation with a positively homogeneous of degree one non-coercive Hamiltonian. The periodic setting was studied earlier by Cardaliaguet, Lions and Souganidis (2009) [16]. Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton–Jacobi equation converge in L∞L∞-weak ⋆ to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media.