Generalized Fokker-Planck equation for the active Ornstein-Uhlenbeck particle

We investigate the dynamics of an inertial active Ornstein-Uhlenbeck particle suspended in a non-Markovian environment. The particle is additionally subjected to external forces, such as harmonic confinement and a magnetic field. Motivated by the importance of understanding the non-Markovian behavior of complex environments, we examine the impact of a viscoelastic medium by employing the Jeffreys fluid framework for modeling the particle motion, which effectively captures both viscous and elastic contributions of the environment. Within this model, we explicitly derive the corresponding Fokker-Planck equation for each case. Building on this, we extend the analysis to the general non-Markovian framework and derive the corresponding generalized Fokker-Planck equation for a free active particle. Furthermore, we obtain the probability distribution function valid for an arbitrary memory kernel under various conditions, including both free and confined motion with and without a magnetic field. This formulation provides a solid basis for analyzing the dynamics of an active particle in a non-Markovian environment, such as mucus and polymer solutions, and further allows the study of relaxation in confined geometries and responses to external fields.