We investigate the stochastic dynamics of an active particle moving at a constant speed under the influence of a fluctuating torque. In our model the angular velocity is generated by a constant torque and random fluctuations described as a Lévy-stable noise. Two situations are investigated.First, we study white Lévy noise where the constant speed and the angular noise generate a persistent motion characterized by the persistence time τ D . At this time scale the crossover from ballistic to normal diffusive behavior is observed. The corresponding diffusion coefficient can be obtained analytically for the whole class of symmetric α-stable noises. As typical for models with noise-driven angular dynamics, the diffusion coefficient depends non-monotonously on the angular noise intensity. As second example, we study angular noise as described by an OrnsteinUhlenbeck process with correlation time τ c driven by the Cauchy white noise. We discuss the asymptotic diffusive properties of this model and obtain the same analytical expression for the diffusion coefficient as in the first case which is thus independent on τ c . Remarkably, for τ c > τ D the crossover from a non-Gaussian to a Gaussian distribution of displacements takes place at a time τ G which can be considerably larger than the persistence time τ D .

We are concerned with the dynamical description of the motion of a stochastic micro-swimmer with constant speed and fluctuating orientation in the long time limit by adiabatic elimination of the orientational variable. Starting with the corresponding full set of Langevin equations, we eliminate the memory in the stochastic orientation and obtain a stochastic equation for the position alone in the overdamped limit. An equivalent procedure based on the Fokker-Planck equation is presented as well.

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