Non-linear Brownian motion: the problem of obtaining the thermal Langevin equation for a non-Gaussian bath

The non-linear dissipation corresponding to a non-Gaussian thermal bath is introduced together with a multiplicative white noise source in the phenomenological Langevin description for the velocity of a particle moving in some potential landscape. Deriving the closed Kolmogorov's equation for the joint probability distribution of the particle displacement and its velocity by use of functional methods and taking into account the well-known Gibbs form of the thermal equilibrium distribution and the condition of 'detailed balance' symmetry, we obtain the exact master equation: given the white noise statistics, this master equation relates the non-linear friction function to the velocity-dependent noise function. In particular, for multiplicative Gaussian white noise this operator equation yields a unique inter-relation between the generally non-linear friction and the (multiplicative) velocity-dependent noise amplitude. This relation allows us to find, for example, the form of velocity-dependent noise function for the case of non-linear Coulomb friction.