Harmonic potential driven by long-range correlated noise.
The probability distribution of a particle in a quadratic potential driven by Gaussian long-range correlated noise has been obtained. The long-time asymptotic relaxation of the stochastic process has been characterized in terms of the long-range correlated noise appearing in the corresponding stochastic differential equation defining the process. The particular case when the particle is driven by a Gaussian color noise has also been revisited, in this way giving its exact probability distribution for all time. By using a characteristic functional technique reported previously, results for a non-Gaussian long-range correlated noise are also shown. Emphasis has been placed on solving the plane rotator in the presence of arbitrary random torques having long-range correlations. In order to show strong and weak non-Markovian effects coming from different sources of noise, the undamped free particle under the influence of arbitrary accelerations has been analyzed. We also analyze in detail the structure of the trajectories of the overdamped and undamped free particle.