We study the effect of a disordered or fractal environment in the irreversible dynamics of a harmonic oscillator. Starting from a generalized Langevin equation and using Laplace analysis, we derive exact expressions for the mean values, variances, and velocity autocorrelation function of the particle in terms of generalized Mittag-Leffler functions. The long-time behaviors of these quantities are obtained and the presence of a whip-back effect is analyzed.

We introduce a Mittag-Leffler correlated random force leading to anomalous diffusion. Starting from a generalized Langevin equation, and using Laplace analysis we derive exact expressions for the mean values, variances and diffusion coefficient for a free particle in terms of generalized Mittag-Leffler functions and its derivatives. The asymptotic behavior of these quantities are obtained, from which the anomalous diffusion behavior of the particle is displayed.

A theoretical framework for analyzing stochastic data from single-particle tracking in viscoelastic materials and under the influence of a trapping potential is presented. Starting from a generalized Langevin equation, we found analytical expressions for the two-time dynamics of a particle subjected to a harmonic potential. The mean-square displacement and the velocity autocorrelation function of the diffusing particle are given in terms of the time lag. In particular, we investigate the subdiffusive case. Using a power-law memory kernel, exact expressions for the mean-square displacement and the velocity autocorrelation function are obtained in terms of Mittag-Leffler functions and their derivatives. The behaviors for short-, intermediate-, and long-time lags are investigated in terms of the involved parameters. Finally, the validity of usual approximations is examined.