Two-dimensional telegraphic processes and their fractional generalization

Abstract

We study the planar motion of telegraphic processes. We derive the two-dimensional telegrapher's equation for isotropic and uniform motions starting from a random walk model which is the two-dimensional version of the multistate random walk with a continuum number of states representing the spatial directions. We generalize the isotropic model and the telegrapher's equation to include planar fractional motions. Earlier, we worked with the one-dimensional version [Masoliver, Phys. Rev. E 93, 052107 (2016)] and derived the three-dimensional version [Masoliver, Phys. Rev. E 96, 022101 (2017)]. An important lesson is that we cannot obtain the two-dimensional version from the three-dimensional or the one-dimensional one from the two-dimensional result. Each dimension must be approached starting from an appropriate random walk model for that dimension.