  We consider an active Brownian particle in a $d$-dimensional harmonic trap,
in the presence of translational diffusion. While the Fokker-Planck equation
can not in general be solved to obtain a closed form solution of the joint
distribution of positions and orientations, as we show, it can be utilized to
evaluate the exact time dependence of all moments, using a Laplace transform
approach. We present explicit calculation of several such moments at arbitrary
times and their evolution to the steady state. In particular we compute the
kurtosis of the displacement, a quantity which clearly shows the difference of
the active steady state properties from the equilibrium Gaussian form. We find
that it increases with activity to asymptotic saturation, but varies
non-monotonically with the trap-stiffness, thereby capturing a recently
observed active- to- passive re-entrant behavior.
