Functional characterization of generalized Langevin equations

TL;DR

This paper develops an exact functional formalism for linear Langevin equations with arbitrary memory kernels and noise, enabling comprehensive analysis of non-Markovian stochastic processes without assuming fluctuation-dissipation relations.

Contribution

It introduces a novel functional approach to analyze generalized Langevin equations with arbitrary noise and memory, extending beyond traditional assumptions.

Findings

Derived an explicit expression for the characteristic functional of the process.

Demonstrated how to compute the Kolmogorov hierarchy for non-Markov processes.

Identified conditions for non-Gaussian statistics and long-tailed distributions.

Abstract

We present an exact functional formalism to deal with linear Langevin equations with arbitrary memory kernels and driven by any noise structure characterized through its characteristic functional. No others hypothesis are assumed over the noise, neither the fluctuation dissipation theorem. We found that the characteristic functional of the linear process can be expressed in terms of noise's functional and the Green function of the deterministic (memory-like) dissipative dynamics. This object allow us to get a procedure to calculate all the 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 analyzed. The introduction of arbitrary fluctuations in fractional Langevin equations have also been pointed out.