It is shown that by splitting the velocity of a diffusing harmonic oscillator into a deterministic spatially dependent part, plus a fluctuating component with zero mean, two basic diffusion operators result in correspondence to the two singular solutions of the related Hamilton-Jacobi-Yasue equation. The diffusion equations with time dependent coefficients which had been proposed before by different authors, are shown to result as linear combinations of these. It is proven the connection of the asymptotic propagator with the two-time transition probability density, in relation to the boundary conditions which are imposed to the system.