  Persistent random walks are intermediate transport processes between a
uniform rectilinear motion and a Brownian motion. They are formed by successive
steps of random finite lengths and directions travelled at a fixed speed. The
isotropic and symmetric one-dimensional persistent random walk is governed by
the telegrapher's equation, also called hyperbolic heat conduction equation.
These equations have been designed to resolve the paradox of the infinite speed
in the heat and diffusion equations. The finiteness of both the speed and the
correlation length leads to several classes of random walks: Persistent random
walk in one dimension can display anomalies that cannot arise for Brownian
motion such as anisotropy and asymmetries. In this work we focus on the case
where the mean free path is anisotropic, the only anomaly leading to a physics
that is different from the telegrapher's case. We derive exact expression of
its Green's function, for its scattering statistics and distribution of
first-passage time at the origin. The phenomenology of the latter shows a
transition for quantities like the escape probability and the residence time.
(Second version that corrects the typos of the first version).
