Lattice random walk. An old problem with a future ahead

This article analyzes the hydrodynamic (continuous) limits of lattice random walks in one spatial dimension. It is shown that a continuous formulation of the process leads naturally to a hyperbolic transport model, characterized by finite propagation velocity, while the classical parabolic limit corresponds to the Kac limit of the hyperbolic model itself. This apparently elementary problem leads to fundamental issues in the theory of stochastic processes and non-equilibrium phenomena, paving the way to new approaches in the field.