SPDEs - a rough path perspective

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We develop the mathematical foundations for an existence and uniqueness theory for rough path driven PDEs in the variational framework. In particular, our approach will rely on a suitable extension of the notion of a variational solution to the rough path case, innovative techniques of estimation of rough integrals and the subsequent derivation of basic energy estimates. The methods that will be developed for these purposes will then be applied to study scalar conservation laws and a related class of degenerate parabolic partial differential equations subject to rough perturbation. More precisely, pursuing the so-called kinetic approach we aim at deriving a corresponding notion of kinetic solution and developing a well-posedness theory for these solutions.