  We study the two-dimensional overdamped motion of an active particle whose
orientational dynamics is subject to fractional Brownian noise, whereas its
position is affected by self-propulsion and Brownian fluctuations. From a
Langevin-like model of active motion with constant swimming speed, we derive
the corresponding Fokker-Planck equation, from which we find the angular
probability density of the particle orientation for arbitrary values of the
Hurst exponent that characterizes the fractional rotational noise. We provide
analytical expressions for the velocity autocorrelation function and the
translational mean-squared displacement, which show that active diffusion
effectively emerges in the long-time limit for all values of the Hurst
exponent. The corresponding expressions for the active diffusion coefficient
and the effective rotational diffusion time are also derived. Our results are
compared with numerical simulations of active particles with rotational motion
driven by fractional Brownian noise, with which we find an excellent agreement.
