  We consider a stochastic conservation law on the line with solution-dependent
diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth,
spatially-homogeneous kick-type random forcing. We show that this Markov
process admits a unique ergodic spatially-homogeneous invariant measure for
each mean in a non-explicit unbounded set. This generalizes previous work on
the stochastic Burgers equation.
