Evolution of a BH embedded in an NFW DM halo, with and without a central stellar bulge, as simulated using the OTIS code. The first three columns correspond to BH trajectories in an NFW halo with initial distance of 20 kpc from the halo centre and orbit ellipticities e = 0, e = 0.5, e = 0.8, while the fourth column includes a central bulge and shows the trajectory of the BH initially at 20 kpc and placed on a circular orbit. In the first and second rows, the trajectory color-code refers to the time elapsed since the beginning of the simulation, as indicated by the color map. Top row: BH trajectories when it is embedded within the static gravitational potential generated by the mass distribution, thus neglecting collisional effects from two-body encounters. Middle row: BH trajectories when collisional effects are included through the analytical description of the DF force (see Eq. (12)). Bottom row: time evolution of the BH’s radial distance from the halo centre. The green line shows the distance when a_BH follows Eq. (5) (or Eq. (22) in the last column) mimicing the effect of a BH embedded in a collisionless medium. The purple line adds the DF contribution given by Eq. (12) for eccentricity e = 0, while the distance corresponding to the middle row of each panel is displayed by the orange curve.