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Under an hypothesis of non-degeneracy of the flux, we study the long-timebehaviour of periodic scalar first-order conservation laws with stochasticforcing in any space dimension. For sub-cubic fluxes, we show the existence ofan invariant measure. Moreover for sub-quadratic fluxes we show uniqueness andergodicity of the invariant measure. Also, since this invariant measure issupported by Lp for some p small, we are led to generalize to the stochasticcase the theory of L1 solutions developed by Chen and Perthame in 2003.

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