Causality, time-reversal invariance and the Langevin equation

It is shown that the assumptions of causality and time-reversal invariance severely restrict the possibility to describe the fluctuations of a variable in a nonlinear Markovian system using a Langevin equation. In fact a theorem is proven which implies that a Langevin force which is independent of the state of the system is necessarily Gaussian and white. The theorem furthermore implies that such a description is only possible if the so-called “systematic force” is proportional to the derivative of the logarithm of the equilibrium distribution of the variable. Our analysis is given for a system with one variable, which may be either even or odd under time reversal.