  This article analyzes the formulation of space-time continuous hyperbolic
hydrodynamic models for systems of interacting particles moving on a lattice,
by connecting their local stochastic lattice dynamics to the formulation of an
associated (space-time continuous) Generalized Poisson-Kac process possessing
the same local transition rules. The hyperbolic hydrodynamic limit follows
naturally from the statistical description of the latter in terms of the system
of its partial probability density functions. Several cases are treated, with
particular attention to: (i) models of interacting particles satisfying an
exclusion principle, and (ii) models defined by a given interparticle
interaction potential. In both cases, the hydrodynamic models may display
singularities, dynamic phase-transitions and bifurcations (as regards the
flux/concentration-gradient constitutive equations), whenever the Kac limit of
the model (infinite propagation velocity limit) is considered.
