  We consider the motion of an active Brownian particle with speed fluctuations
in d-dimensions in the presence of both translational and orientational
diffusion. We use an Ornstein-Uhlenbeck process for active speed generation.
Using a Laplace transform approach, we describe and use a Fokker-Planck
equation-based method to evaluate the exact time dependence of all relevant
dynamical moments. We present explicit calculations of such moments and compare
our analytical predictions against numerical simulations to demonstrate and
analyze several dynamical crossovers. The kurtosis of displacement shows
positive or negative deviations from a Gaussian behavior at intermediate times
depending on the dominance of speed or orientational fluctuations.
