[outline of the report]

      As a prelude to the results of the following research project, a general view of stochastic processes (including Wiener, Brownian and Ornstein-Uhlenbeck ones), their spectral properties and the characterisation of their (ir)-reversibility is set out. We briefly recall the Arrhenius-Kramers escape problem for passive particles evolving between two metastable states of a uni-dimensional potential. The Active Ornstein-Uhlenbeck Process, for which reversible and Markovian properties are discussed in depth, is finally introduce.

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  Since active Ornstein-Uhlenbeck particles, i.e. overdamped particles evolving in a potential and subject to a noisy self-propulsion velocity, can be mapped into a system of underdamped particles under additional velocity dependent forces modeled by an Ornstein-Uhlenbeck process and subject to a white noise, one is able to provide a Fokker-Planck-Kolmogorov based analysis in a low temperature regime. By resorting to a very first idea developed by Kłosek-Dygas, Matkowsky and Schuss, one can build a generic procedure allowing us to compute analytically each coefficient of the perturbative expansion of the stationary distribution in a limit of weak activity, i.e. weak temporal correlations of the noise. Motivated by the concordance between our results and the pre-existing ones derived by paths integrals methods, we develop a controlled extension of the weak activity regime based on Borel-Laplace resummation procedure for divergent series, in view of reaching the strong activity limit and understanding the expected undergoing phase transition.

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    In the weak activity regime, we attend to construct a well controlled extension of the uni-dimensional procedure to the case of a bi-dimensional active Ornstein-Uhlenbeck process, providing us with the two first corrections to the passive Boltzmann-Gibbs stationary distribution. The ensuring analytical results allow us to predict new phenomena due to the existing non-local correction of the stationary probability in the presence of activity : a trapping effect (localization by non-locality) and a ratchet effect (ratchet by transverse non-locality). The laters are cross-checked with numerical simulations.