Persistent random walkers and the telegrapher's equation

Persistent random walkers (who remember the direction of their last step)
are currently commonly used to model run and tumble bacteria. However they have been studied for some time before  this. I'll review some simple results on the
relaxation time for a single persistent random walker which show that it can be
faster than the usual diffusive scaling relaxation time i.e $ \tau \sim O(L)$
rather than $\tau \sim O(L^2)$. I'll also show how a continuum limit for the Master equation yields the telegrapher's equation which is an equation that interpolates between the diffusion equation and the wave equation. I'll discuss the solution of the telegraphers equation. If time  permits I'll mention some results derived with  Alex Slowman and Richard Blythe  on two interacting persistent walkers.

References:

Some applications of persistent random walkers and the telegrapher's equation
G. H. Weiss Physica A  311 (2002) p381-410

Jamming and Attraction of Interacting Run-and-Tumble Random Walkers
A.B. Slowman, M.R. Evans, and R.A. Blythe
Phys. Rev. Lett. 116, 218101 (2016)