Some remarks on the adiabatic elimination of fast variables from coupled Langevin equations

Recently we discussed the adiabatic elimination of fast variables from a many-dimensional Fokker-Planck equation linear in the fast variables. Here we show how the same results can be obtained by manipulation of the equivalent set of coupled Langevin equations. At first sight the Langevin formalism has the advantage of greater transparency. However, we found it necessary to go back to the Fokker-Planck formalism to resolve the ambiguity in the interpretation of the nonlinear Langevin equations resulting from the elimination procedure (Itô-Stratonovich ambiguity).