The equations of motion for a classical newtonian system subjected to a random additive force and a viscous resistance proportional to velocity can be described according to the equation of Hamilton and Jacobi, subsequently modified by Yasue and others in order to include viscosity effects. The present approach proposes a partitioning of the solutions to that equation so as to establish a direct correspondence between the components of the solution and the various terms of a diffusion operator acting upon a probability density in configuration space, thus leading to interpretation of different physical effects. The velocity is separated into an Eulerian plus a Lagrangian component, and this splitting is dependent upon an arbitrary function in the Hamilton-Jacobi-Yasue equation. Consequently, the Eulerian component originates the drift term of diffusion, while the Lagrangian one yields the diffusive terms, and the arbitrary separating function is adjusted so as to put the memory term equal to zero. This proves to be possible in most cases of interest in an almost exact way. As a consequence, the drift component verifies an equation of Riccati, modified so as to include frictional terms.

The procedure is applicable without modification to systems driven by colored noise and position-dependent friction.

Applications have been done to unidimensional Brownian motion in the limit of high friction, thus a fourth-order differential equation has been obtained describing diffusion in configuration space, which is exact up to the inverse seventh power of the viscosity coefficient. This equation has been obtained directly in configuration space, by functional-integral techniques, thus avoiding the complicated procedures which are necessary in order to project the diffusion process in phase-space onto the configurational subspace.

The boundary conditions strongly influence the diffusion process even in the limit of large time, and the choice of different solutions of the Riccati equation corresponding to different boundary conditions leads to descriptions of the process, which govern fluctuations belonging to different time scales. However, the identification of the asymptotic propagator with the two-time transition probability density meets with difficulties, being founded upon the Markovian approximation to the process. This approximation has been analysed in detail for the harmonic oscillator.

The consideration of systems driven by colored noise has been presently restricted mainly to the motion of a charged particle in an electromagnetic random field. In an electromagnetic zero-point-field, it results that for a class of systems, including the harmonic oscillator, the diffusion equation and the Riccati equation both result to be equivalent to an equation of Schrödinger. The calculation provides also an expression for the nonrelativistic Lamb-shifts which result to exhibit the correct dependence upon the physical parameters.