  We solve the time-dependent Fokker-Planck equation for a two-dimensional
active Brownian particle exploring a circular region with an absorbing
boundary. Using the passive Brownian particle as basis states and dealing with
the activity as a perturbation, we provide a matrix representation of the
Fokker-Planck operator and we express the propagator in terms of the perturbed
eigenvalues and eigenfunctions. Alternatively, we show that the propagator can
be expressed as a combination of the equilibrium eigenstates with weights
depending only on time and on the initial conditions, and obeying exact
iterative relations. Our solution allows also obtaining the survival
probability and the first-passage time distribution. These latter quantities
exhibit peculiarities induced by the non-equilibrium character of the dynamics,
in particular, they display a strong dependence on the activity of the particle
and, to a less extent, also on its rotational diffusivity.
