Langevin equation for a free particle driven by power law type of noises

• We study a generalized Langevin equation driven by mixture of internal noises.
• We examine the mean square displacement and velocity autocorrelation function.
• The multinomial and three parameter Mittag-Leffler functions play an important role in the theory of anomalous diffusion.

We study a generalized Langevin equation for a free particle driven by N internal noises. The mean square displacement and velocity autocorrelation function are derived in case of a mixture of Dirac delta, power law and Mittag-Leffler noises. Additionally, a frictional memory kernel of distributed order is considered. The long time limit and short time limit are analyzed, and the dominant contributions of noises on particle dynamics is discussed. Various different diffusive behaviors (subdiffusion, superdiffusion, normal diffusion, ultraslow diffusion) are obtained. The considered problem may be used in the theory of anomalous diffusion in complex environment.