Quantum behavior of a classical particle subject to a random force

We give a partial answer to the question whether the Schrodinger equation can be derived from the Newtonian mechanics of a particle in a potential subject to a random force. We show that the fluctuations around the classical motion of a one dimensional harmonic oscillator subject to a small random force can be described by the Schrodinger equation for a period of time depending on the frequency and the energy of the oscillator. We achieve this by deriving the postulates of Nelson's stochastic formulation of quantum mechanics for sufficiently small random forces. We show that the same result applies to small potential perturbations around the harmonic oscillator as long as the total potential preserves the periodicity of motion with a small shift in frequency. We discuss heuristics to generalize the result for a particle in one dimension in a potential where the motion can be described using action-angle variables.