Generalized Langevin equation with a three parameter Mittag-Leffler noise

The relaxation functions for a given generalized Langevin equation in the presence of a three parameter Mittag-Leffler noise are studied analytically. The results are represented by three parameter Mittag-Leffler functions. Exact results for the velocity and displacement correlation functions of a diffusing particle are obtained by using the Laplace transform method. The asymptotic behavior of the particle in the short and long time limits are found by using the Tauberian theorems. It is shown that for large times the particle motion is subdiffusive for β − 1 < α δ < β , and superdiffusive for β < α δ . Many previously obtained results are recovered. Due to the many parameters contained in the noise term, the model considered in this work may be used to improve the description of data and to model anomalous diffusive processes in complex media.