  The study of diffusion with preferential returns to places visited in the
past has attracted an increased attention in recent years. In these highly
non-Markov processes, a standard diffusive particle intermittently resets at a
given rate to previously visited positions. At each reset, a position to be
revisited is randomly chosen with a probability proportional to the accumulated
amount of time spent by the particle at that position. These preferential
revisits typically generate a very slow diffusion, logarithmic in time, but
still with a Gaussian position distribution at late times. Here we consider an
active version of this model, where between resets the particle is
self-propelled with constant speed and switches direction in one dimension
according to a telegraphic noise. Hence there are two sources of
non-Markovianity in the problem. We exactly derive the position distribution in
Fourier space, as well as the variance of the position at all times. The
crossover from the short-time ballistic regime, dominated by activity, to the
large-time anomalous logarithmic growth induced by memory is studied. We also
analytically derive a large deviation principle for the position, which
exhibits a logarithmic time-scaling instead of the usual algebraic form.
Interestingly, at large distances, the large deviations become independent of
time and match the non-equilibrium steady state of a particle under resetting
to its starting position only.
