Large-time behavior for Hamilton-Jacobi equations forced by additive noise

We consider Hamilton-Jacobi equations with or without viscosity with periodic boundary conditions under the influence of additive noise, white in time and periodic and twice differentiable in space. We show for certain Hamiltonians the existence of a solution global in time, which is unique up to constants and attracts up to constants any other solution. This solution can be used to construct measures which are invariant under the evolution. We assume superlinear growth, but no convexity of the Hamiltonian.