A stochastic model related to the telegrapher's equation

We will consider a very simple stochastic model, a random walk. Unfortunately, this model is little known. It has very interesting features and leads not to a diffusion equation but to a hyperbolic one. The model first appeared in the literature in a paper by Sidney Goldstein, known to you mostly because of his work in fluid dynamics. The model had first been proposed by G. I. Taylor — I think in an abortive, or at least not very successful, attempt to treat turbulent diffusion. But the model itself proved to be very interesting. The problem is the following: Suppose you have a lattice of points. I mean discrete, equidistant points as in Figure 1.