Active Brownian motion in two dimensions

We study the dynamics of a single active Brownian particle (ABP) in two spatial dimensions. The ABP has an intrinsic timescale DR^-1 set by the rotational diffusion constant DR. We show that, at short times tDR^-1, the presence of ``activeness'' results in a strongly anisotropic and nondiffusive dynamics in the $(xy)$ plane. We compute exactly the marginal distributions of the x and y position coordinates along with the radial distribution, which are all shown to be non-Brownian. In addition, we show that, at early times, the ABP has anomalous first-passage properties, characterized by non-Brownian exponents.