Reduced Markovian descriptions of the Brownian oscillator. Towards an exact theory

We discuss the derivation of a reduction scheme for a harmonically bound Brownian particle in the overdamped regime, which allows to encompass and generalize the Smoluchowski equation. We derive a contracted description which complies with the prescriptions of the Invariant Manifold theory. The drift coefficient of the reduced dynamics is obtained via an exact summation of the Chapman-Enskog expansion. The structure of the diffusion coefficient becomes clear after establishing the Fluctuation-Dissipation Theorem. Our study paves the way to the development of model reduction procedures applicable to more general diffusion processes subject to non-linear interactions.