Abstract
The action function for a one-dimensional particle in a dissipative medium is obtained in a formal way by splitting the Hamilton-Jacobi equation in a deterministic and a stochastic part. For an harmonic oscillator, the velocity turns out to be given by a linear Langevin equation with coloured Gaussian noise. In the overdamped case, there appear two types of solutions to the HJ equation, one which describes particles mainly controlled by their kinetic energy, the other by the drift velocity, due to the potential. These lead to different configurational diffusion equations. A more general analysis of the solutions expanded in powers of the inverse of the viscosity coefficient is carried out for any potential and similar results are obtained.
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