Functional characterization of generalized Langevin equations

Abstract

We present an exact functional formalism to deal with linear Langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional. No other hypothesis is assumed over the noise, neither do we use the fluctuation–dissipation theorem. We find that the characteristic functional of the linear process can be expressed in terms of noise functional and the Green function of the deterministic (memory-like) dissipative dynamics. This yields a procedure for calculating the full Kolmogorov hierarchy of the non-Markov process. As examples, we have characterized through the 1-time probability a noise-induced interplay between the dissipative dynamics and the structure of different noises. Conditions that lead to non-Gaussian statistics and distributions with long tails are analysed. The introduction of arbitrary fluctuations in fractional Langevin equations has also been pointed out.