Path-dependent convex conservation laws

For scalar conservation laws driven by a rough path $z(t)$, in the sense of Lions, Perthame and Souganidis in arXiv:1309.1931, we show that it is possible to replace $z(t)$ by a piecewise linear path, and still obtain the same solution at a given time, under the assumption of a convex flux function in one spatial dimension. This result is connected to the spatial regularity of solutions. We show that solutions are spatially Lipschitz continuous for a given set of times, depending on the path and the initial data. Fine properties of the map $z \mapsto u(\tau)$, for a fixed time $\tau$, are studied. We provide a detailed description of the properties of the rough path $z(t)$ that influences the solution. This description is extracted by a "factorization" of the solution operator (at time $\tau$). In a companion paper, we make use of the observations herein to construct computationally efficient numerical methods.