Stochastic Langevin equations: Markovian and non-Markovian dynamics.

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 detail. 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.