Evolution of the semimajor axis (top), the eccentricity (middle), and the inclination (bottom) for an inclined (ι₀ = 0.5°) and eccentric (e₀ = 0.1) BH (m_BH = 20 M_⊙) undergoing dynamical friction in the AGN disk. The thick blue line indicates the evolution without the turbulent velocity field, while each thin red line includes a different realization of the turbulent velocity field. The BH is placed at the migration trap within an AGN disk around a 10⁷ M_⊙ SMBH. The dotted black line is the mean evolution of the realizations including turbulence. In the top panel, the dot-dashed lines indicate the mean square change in semimajor axis ⟨Δa²⟩ (Eq. (44)). In the middle and bottom panels, the dot-dashed green line is the ratio between the turbulent velocity dispersion σ_turb and the circular velocity v_circ. The vertical dashed lines indicate the first three midplane crossings. The turbulence prevents the full circularization and alignment of the embedded BH.