We introduce a perturbative method to calculate all moments of the first passage time distribution in stochastic one-dimensional processes which are subject to both white and colored noise. This class of non-Markovian processes is at the center of the study of thermal active matter, that is self-propelled particles subject to diffusion. The perturbation theory about the Markov process considers the effect of self-propulsion to be small compared to that of thermal fluctuations. To illustrate our method, we apply it to the case of active thermal particles (i) in a harmonic trap and (ii) on a ring. For both we calculate the first-order correction of the moment-generating function of first passage times, and thus to all its moments. Our analytical results are compared to numerics.
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