Hamilton–Jacobi and Fokker–Planck equations for the harmonic oscillator

Using Feynman’s path integral formalism applied to stochastic classical processes, we show the equivalence between the Hamilton–Jacobi (HJ) and Fokker–Planck (FP) equations, associated with an overdamped Brownian harmonic oscillator. In this case, the Langevin equation leads to a Gaussian Lagrangian function and then the path integration which defines the conditional probability density can be replaced by the extremal path. Due to this fact and following the classical dynamics formalism, we prove the strict equivalence between the HJ and FP equations. We do this first for an ordinary Brownian harmonic oscillator and then the proof is extended to an electrically charged Brownian particle under the action of force fields: magnetic field and additional time-dependent force fields. We observe that this extremal action principle allows us to derive in a straightforward way not only the HJ differential equation, but also its solution, the extremal action.