  Nonlinear, multiplicative Langevin equations for a complete set of slow
variables in equilibrium systems are generally derived on the basis of the
separation of time scales. The form of the equations is universal and
equivalent to that obtained by Green. An equation with a nonlinear friction
term for Brownian motion turns out to be an example of the general results. A
key method in our derivation is to use different discretization schemes in a
path integral formulation and the corresponding Langevin equation, which also
leads to a consistent understanding of apparently different expressions for the
path integral in previous studies.
