  By working in the small persistence time limit, we determine the steady-state
distribution of an Active Ornstein Uhlenbeck Particle (AOUP) experiencing, in
addition to self-propulsion, a Gaussian white noise modelling a bath at
temperature T. This allows us to derive analytical formulas for three
quantities: the spatial density of a confined particle, the current induced by
an asymmetric periodic potential and the entropy production rate. These
formulas disentangle the respective roles of the passive and active noises on
the steady state of AOUPs, showing that signatures of non-equilibrium can
display surprising behaviors as the temperature is varied. Indeed, depending on
the potential in which the particle evolves, both the current and the entropy
production rate can be non-monotonic functions of T. The latter can even
diverge at high temperature for steep enough confining potentials. Thus,
depending on context, switching on translational diffusion may drive the
particle closer to or further away from equilibrium. We then probe the range of
validity of our quantitative derivations by numerical simulations. Finally, we
explain how the method presented here to tackle perturbatively an Ornstein
Uhlenbeck (OU) noise could be further generalized beyond the Brownian case.
