The persistent random walk is a modification of the random walk model.
A population of particles are distributed on a line, with constant speed
, and each particle's velocity may be reversed at any moment.
The reversal time is exponentially distributed as
, then the population density
{\displaystyle (2\tau ^{-1}\partial _{t}+\partial _{tt}-c_{0}^{2}\partial _{xx})n=0}
which is the telegrapher's equation.