Broad class of nonlinear Langevin equations driven by multiplicative Ornstein–Uhlenbeck noise: Generalized n-moment and generalized second Einstein relation

A wide class of the nonlinear Langevin equations driven by a generic multiplicative Ornstein–Uhlenbeck noise is investigated, with time–space-dependent drift and diffusion coefficients. Solutions for the probability density function and generalized n-moment (averages of a generic function) are obtained. Also, a generalized second Einstein relation is attained. These generalizations lead to novel associations among certain characteristics of the stochastic systems. Given the broad applicability of these models, the present general theoretical results might be useful in the discussion of different processes.