  Mott variable range hopping is a fundamental mechanism for low-temperature
electron conduction in disordered solids in the regime of Anderson
localization. In a mean field approximation, it reduces to a random walk
(shortly, Mott random walk) on a random marked point process with possible
long-range jumps. We consider here the one-dimensional Mott random walk and we
add an external field (or a bias to the right). We show that the bias makes the
walk transient, and investigate its linear speed. Our main results are
conditions for ballisticity (positive linear speed) and for sub-ballisticity
(zero linear speed), and the existence in the ballistic regime of an invariant
distribution for the environment viewed from the walker, which is mutually
absolutely continuous with respect to the original law of the environment. If
the point process is a renewal process, the aforementioned conditions result in
a sharp criterion for ballisticity. Interestingly, the speed is not always
continuous as a function of the bias.
