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 $$L^p$$Lp for some $$p$$p small, we are led to generalize to the stochastic case the theory of $$L^1$$L1 solutions developed by Chen and Perthame (Ann Inst H Poincaré Anal Non Linéaire 20(4):645–668, 2003).

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