A {\em propagation-dispersion equation} is derived for the first passage
distribution function of a particle moving on a substrate with time delays. The
equation is obtained as the continuous limit of the {\em first visit equation},
an exact microscopic finite difference equation describing the motion of a
particle on a lattice whose sites operate as {\em time-delayers}. The
propagation-dispersion equation should be contrasted with the
advection-diffusion equation (or the classical Fokker-Planck equation) as it
describes a dispersion process in {\em time} (instead of diffusion in space)
with a drift expressed by a propagation speed with non-zero bounded values. The
{\em temporal dispersion} coefficient is shown to exhibit a form analogous to
Taylor's dispersivity. Physical systems where the propagation-dispersion
equation applies are discussed.Comment: 12 pages+ 5 figures, revised and extended versio