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.
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