Persistent random walk

The persistent random walk is a modification of the random walk model.

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