  We compare the fluctuations in the velocity and in the fraction of time spent
at a given position for minimal models of a passive and an active particle: an
asymmetric random walker and a run-and-tumble particle in continuous time and
on a 1D lattice. We compute rate functions and effective dynamics conditioned
on large deviations for these observables. While generally different, for a
unique and non-trivial choice of rates (up to a rescaling of time) the velocity
rate functions for the two models become identical, whereas the effective
processes generating the fluctuations remain distinct. This equivalence
coincides with a remarkable parity of the spectra of the processes' generators.
For the occupation-time problem, we show that both the passive and active
particles undergo a prototypical dynamical phase transition when the average
velocity is non-vanishing in the long-time limit.
