Invariant measure of scalar first-order conservation laws with stochastic forcing

Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour of periodic scalar first-order conservation laws with stochastic forcing in any space dimension. For sub-cubic fluxes, we show the existence of an invariant measure. Moreover for sub-quadratic fluxes we show uniqueness and ergodicity of the invariant measure. Also, since this invariant measure is supported by for some small, we are led to generalize to the stochastic case the theory of solutions developed by Chen and Perthame (Ann Inst H Poincar, Anal Non Lin,aire 20(4):645-668, 2003).