Space-time transport schemes and homogenization. II. Extension of the theory and applications

The theory of space-time and non-Markovian spin-driven Diusion processes introduced in Giona (2016 J. Stat. Mech. submitted) is extended to random space-time displacements and to a continuum of spins. The comparison of space-time diusion models with the continuous time random walk of Montroll and Weiss is developed in order to highlight the analogies and the dierences between these two processes. A similar comparison is performed for other classes of processes such as the multistate random walk and the correlated continuous time random walk. Moreover, the article develops the correspondence between space-time and non-Markovian spin-driven diffusion models, analyzes their relativistic properties, and introduces the concept of hyperbolic homogenization. Some preliminary observations on the use of these models in order to frame relativistic quantum mechanics within a stochastic transport paradigm without enforcing the analytic continuation of the time variable to the imaginary axis are outlined.