  Brownian dynamics of a self-propelled particle in linear shear flow is
studied analytically by solving the Langevin equation and in simulation. The
particle has a constant propagation speed along a fluctuating orientation and
is additionally subjected to a constant torque. In two spatial dimensions, the
mean trajectory and the mean square displacement (MSD) are calculated as
functions of time t analytically. In general, the mean trajectories are
cycloids that are modified by finite temperature effects. With regard to the
MSD different regimes are identified where the MSD scales with t^a with a =
0,1,2,3,4. In particular, an accelerated (a = 4) motion emerges if the particle
is self-propelled along the gradient direction of the shear flow.
