Diffusion in configuration space according to the choice of the solution of the Hamilton-Jacobi-Yasue equation and the role of boundary conditions

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. (C) 1996 American Institute of Physics.