Stochastic Langevin equations: Markovian and non-Markovian dynamics

TL;DR

This paper compares non-Markovian and Markovian Langevin equations, analyzing the validity of local approximations in systems with additive and multiplicative noise, and explicitly specifies conditions for their accuracy.

Contribution

It provides a detailed analysis of when local Markovian approximations are valid for non-Markovian Langevin equations with various noise types.

Findings

Local approximation validity depends on specific system parameters.

Explicit conditions for good approximation are derived.

Analysis includes both additive and multiplicative noise cases.

Abstract

Non-Markovian stochastic Langevin-like equations of motion are compared to their corresponding Markovian (local) approximations. The validity of the local approximation for these equations, when contrasted with the fully nonlocal ones, is analyzed in details. The conditions for when the equation in a local form can be considered a good approximation are then explicitly specified. We study both the cases of additive and multiplicative noises, including system dependent dissipation terms, according to the Fluctuation-Dissipation theorem.