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We demonstrate that a Langevin equation that describes the motion of aBrownian particle under non-equilibrium conditions can be exactly transformedto a special equation that explicitly exhibits the response of the velocity toa time dependent perturbation. This transformation is constructed on the basisof an operator formulation originally used in nonlinear perturbation theory fordifferential equations by extending it to stochastic analysis. We find that theobtained expression is useful for the calculation of fundamental quantities ofthe system, and that it provides a physical basis for the decomposition of theforces in the Langevin description into effective driving, dissipative, andrandom forces in a large-scale description.Comment: 14 pages, to appear in J. Phys. A: Math. Ge

Exact transformation of a Langevin equation to a fluctuating response equation