  Overdamped Brownian motion of a self-propelled particle is studied by solving
the Langevin equation analytically. On top of translational and rotational
diffusion, in the context of the presented model, the "active" particle is
driven along its internal orientation axis. We calculate the first four moments
of the probability distribution function for displacements as a function of
time for a spherical particle with isotropic translational diffusion as well as
for an anisotropic ellipsoidal particle. In both cases the translational and
rotational motion is either unconfined or confined to one or two dimensions. A
significant non-Gaussian behavior at finite times t is signalled by a
non-vanishing kurtosis. To delimit the super-diffusive regime, which occurs at
intermediate times, two time scales are identified. For certain model
situations a characteristic t^3 behavior of the mean square displacement is
observed. Comparing the dynamics of real and artificial microswimmers like
bacteria or catalytically driven Janus particles to our analytical expressions
reveals whether their motion is Brownian or not.
