  We provide an analytical solution for the time-dependent Fokker-Planck
equation for a two-dimensional active Brownian particle trapped in an isotropic
harmonic potential. Using the passive Brownian particle as basis states we show
that the Fokker-Planck operator becomes lower diagonal, implying that the
eigenvalues are unaffected by the activity. The propagator is then expressed as
a combination of the equilibrium eigenstates with weights obeying exact
iterative relations. We show that for the low-order correlation functions, such
as the positional autocorrelation function, the recursion terminates at finite
order in the P\'eclet number allowing us to generate exact compact expressions
and derive the velocity autocorrelation function and the time-dependent
diffusion coefficient. The nonmonotonic behavior of latter quantities serves as
a fingerprint of the non-equilibrium dynamics.
