Scalar conservation laws with rough flux and stochastic forcing

In this paper, we study scalar conservation laws where the flux is driven by a geometric H\"older $p$-rough path for some $p\in (2,3)$ and the forcing is given by an It\^o stochastic integral driven by a Brownian motion. In particular, we derive the corresponding kinetic formulation and define an appropriate notion of kinetic solution. In this context, we are able to establish well-posedness, i.e. existence, uniqueness and the $L^1$-contraction property that leads to continuous dependence on initial condition. Our approach combines tools from rough path analysis, stochastic analysis and theory of kinetic solutions for conservation laws. As an application, this allows to cover the case of flux driven for instance by another (independent) Brownian motion enhanced with L\'evy's stochastic area.