Abstract
Abstract In this work we consider a generalised Ornstein–Uhlenbeck (O–U) process for a stochastically driven particle in an harmonic potential which is governed by a Fokker–Planck equation in the presence of a memory kernel. We analyse the probability density function, the mean and the mean squared displacement (MSD) by employing the subordination approach connecting the operational time of the process with the (generalised) laboratory time. We provide analytical results for the mean and the MSD in case of a power-law memory kernel which corresponds to the fractional O–U process. The generalised O–U process in the presence of Poissonian resetting is also investigated by using the renewal equation approach, and the nonequilibrium stationary state approached in the long time limit is obtained. The analytical results are confirmed by numerical simulations based on the coupled Langevin equations.
Published Version
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