Weak versus $\mathcal{D}$-solutions to linear hyperbolic first order systems with constant coefficients

We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of $\mathcal{D}$-solutions introduced in the recent paper [K8], which arose in connection to vectorial Calculus of Variations in $L^{\mathrm{\infty }}$ and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.