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 ${\displaystyle c_{0}}$, and each particle's velocity may be reversed at any moment. The reversal time is exponentially distributed as ${\displaystyle e^{-t/\tau }/\tau }$, then the population density ${\displaystyle n}$ evolves according to¹$${\displaystyle (2\tau ^{-1}\partial _{t}+\partial _{tt}-c_{0}^{2}\partial _{xx})n=0}$$which is the telegrapher's equation.