Turbulent drag on stellar mass black holes embedded in disks of active galactic nuclei

Context. Interactions between stellar-mass black holes (BHs) and the accretion disks of supermassive BHs in active galactic nuclei (AGN) constitute a promising channel for the formation of gravitational wave sources. The efficiency of this process depends critically on how embedded BHs evolve under the influence of gaseous drag. Previous studies have assumed laminar disk conditions, leading to idealized configurations with BHs on circular, coplanar orbits. However, AGN disks are expected to be turbulent, and the impact of turbulence on BH orbital evolution remains largely unexplored. Aims. We investigate how AGN disk turbulence affects the orbital dynamics of a stellar-mass BH initially located at a migration trap, focusing on the long-term behavior of eccentricity and inclination in the quasi-embedded regime. Methods. We developed a semi-analytical framework in which turbulence is modeled as a stochastic velocity field acting through a modified drag force. We integrated the resulting stochastic differential equations both in Cartesian coordinates and in orbital elements using a linearized perturbative approach and compared these results with full numerical simulations. Results. Eccentricity and inclination evolve toward steady-state Rayleigh distributions, with variances determined by the local disk properties and the ratio of the gas damping rate to the orbital frequency. The analytical predictions agree well with the numerical simulations. We provide closed-form expressions for the variances in both the fast and slow damping regimes. These results are directly applicable to Monte Carlo population models and can serve as physically motivated initial conditions for hydrodynamical simulations. Conclusions. Turbulent forcing prevents full circularization and alignment of BH orbits in AGN disks, even in the presence of strong gas drag. This has important implications for BH merger and binary formation rates, which are sensitive to the residual eccentricity and inclination. Our results highlight the need to account for turbulence-induced stochastic heating when modeling the dynamical evolution of compact objects in AGN environments.