  We revisit the problem of the overdamped (large friction) limit of the
Brownian dynamics in an inhomogeneous medium characterized by a
position-dependent friction coefficient and a multiplicative noise (local
temperature) in one space dimension. Starting from the Kramers equation and
analyzing it through the expansion in terms of eigenfunctions of a quantum
harmonic oscillator, we derive analytically the corresponding Fokker-Planck
equation in the overdamped limit. The result is fully consistent with the
previous finding by Sancho, San Miguel, and D\"urr \cite{Sanc82}. Our method
allows us to generalize the Brinkman's hierarchy, and thus it would be
straightforward to obtain higher-order corrections in a systematic inverse
friction expansion without any assumption. Our results are confirmed by
numerical simulations for simple examples.
