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We investigate the stochastic dynamics of an active particle moving at aconstant speed under the influence of a fluctuating torque. In our model theangular velocity is generated by a constant torque and random fluctuationsdescribed as a L\'evy-stable noise. Two situations are investigated. First, westudy white L\'evy noise where the constant speed and the angular noisegenerate a persistent motion characterized by the persistence time $\tau_D$. Atthis time scale the crossover from ballistic to normal diffusive behavior isobserved. The corresponding diffusion coefficient can be obtained analyticallyfor the whole class of symmetric $\alpha$-stable noises. As typical for modelswith noise-driven angular dynamics, the diffusion coefficient dependsnon-monotonously on the angular noise intensity. As second example, we studyangular noise as described by an Ornstein-Uhlenbeck process with correlationtime $\tau_c$ driven by the Cauchy white noise. We discuss the asymptoticdiffusive properties of this model and obtain the same analytical expressionfor the diffusion coefficient as in the first case which is thus independent on$\tau_c$. Remarkably, for $\tau_c>\tau_D$ the crossover from a non-Gaussian toa Gaussian distribution of displacements takes place at a time $\tau_G$ whichcan be considerably larger than the persistence time $\tau_D$.

Diffusion of Active Particles With Stochastic Torques Modeled as $\alpha$-Stable Noise