  We study the dynamical properties of an active particle subject to a swimming
speed explicitly depending on the particle position. The oscillating spatial
profile of the swim velocity considered in this paper takes inspiration from
experimental studies based on Janus particles whose speed can be modulated by
an external source of light. We suggest and apply an appropriate model of an
active Ornstein Uhlenbeck particle to the present case. This allows us to
predict the stationary properties, by finding the exact solution of the
steady-state probability distribution of particle position and velocity. From
this, we obtain the spatial density profile and show that its form is
consistent with the one found in the framework of other popular models. The
reduced velocity distribution highlights the emergence of non-Gaussianity in
our generalized AOUP model which becomes more evident as the spatial dependence
of the velocity profile becomes more pronounced. Then, we focus on the
time-dependent properties of the system. Velocity autocorrelation functions are
studied in the steady-state combining numerical and analytical methods derived
under suitable approximations. We observe a non-monotonic decay in the temporal
shape of the velocity autocorrelation function which depends on the ratio
between the persistence length and the spatial period of the swim velocity.
Finally, we numerically and analytically study the mean square displacement and
the long-time diffusion coefficient. The ballistic regime, observed in the
small-time region, is deeply affected by the properties of the swim velocity
landscape which induces also a crossover to a sub-ballistic but superdiffusive
regime for intermediate times. Finally, the long-time diffusion coefficient
decreases as the amplitude of the swim velocity oscillations increases because
the diffusion is determined by those regions where the particles are slow.
