  We study the free diffusion in two dimensions of active-Brownian swimmers
subject to passive fluctuations on the translational motion and to active
fluctuations on the rotational one. The Smoluchowski equation is derived from a
Langevin-like model of active swimmers, and analytically solved in the
long-time regime for arbitrary values of the P\'eclet number, this allows us to
analyze the out-of-equilibrium evolution of the positions distribution of
active particles at all time regimes. Explicit expressions for the mean-square
displacement and for the kurtosis of the probability distribution function are
presented, and the effects of persistence discussed. We show through Brownian
dynamics simulations that our prescription for the mean-square displacement
gives the exact time dependence at all times. The departure of the probability
distribution from a Gaussian, measured by the kurtosis, is also analyzed both
analytically and computationally. We find that for P\'eclet numbers $\lesssim
0.1$, the distance from Gaussian increases as $\sim t^{-2}$ at short times,
while it diminishes as $\sim t^{-1}$ in the asymptotic limit.
