  Using the path integral representation of the non-equilibrium dynamics, we
compute the most probable path between arbitrary starting and final points,
followed by an active particle driven by persistent noise. We focus our
attention on the case of active particles immersed in harmonic potentials,
where the trajectory can be computed analytically. Once we consider the
extended Markovian dynamics where the self-propulsive drive evolves according
to an Ornstein-Uhlenbeck process, we can compute the trajectory analytically
with arbitrary conditions on position and self-propulsion velocity. We test the
analytical predictions against numerical simulations and we compare the
analytical results with those obtained within approximated equilibrium-like
dynamics.
