  The conventional equations of Brownian motion can be derived from the first
principles to order $\lambda^2=m/M$, where $m$ and $M$ are the masses of a bath
molecule and a Brownian particle respectively. We discuss the extension to
order $\lambda^4$ using a perturbation analysis of the Kramers-Moyal expansion.
For the momentum distribution such method yields an equation whose stationary
solution is inconsistent with Boltzmann-Gibbs statistics. This property
originates entirely from non-Markovian corrections which are negligible in
lowest order but contribute to order $\lambda^4$.
