Proton energy spectra ϵ² dn_cor/dϵ (per log-interval of energy) vs. proton energy ϵ predicted by stochastic acceleration in a turbulent corona, starting from mono-energetic protons with a Lorentz factor of γ₀ ∼ 1. In each panel, the solid line corresponds to a solution obtained by integrating the Fokker-Planck equation up to time τ_adv = r_co/v_adv (model (1) including energy losses, advection, and escape as described in the text), and the dashed line shows the corresponding solution for the same parameters for the generalized Fermi model described by Eq. (5) (model (2)). In all panels, r_co is set to 30r_g, and in the top and middle panels, v_adv ≃ 0.02 c. The upper panel shows the dependence on v_A, assuming otherwise a coherence length scale for the turbulence ℓ_c = 10 r_g and an advection velocity v_adv = 0.02 c. From left to right, as indicated, v_A/c = 0.05, v_A/c = 0.10, v_A/c = 0.20, and v_A/c = 0.40. The middle panel examines the dependence on ℓ_c, assuming otherwise v_A/c = 0.20 and v_adv/c = 0.02. From right to left, as indicated, ℓ_c = 3 r_g, ℓ_c = 10 r_g, and ℓ_c = 30 r_g. The bottom panel examines the influence of advection, characterized here by the advection velocity v_adv (which sets the advection timescale τ_adv ≡ r_co/v_adv), assuming otherwise v_A/c = 0.20 and ℓ_c = 3 r_g. From right to left, as indicated, v_adv/c = 0.02, v_adv/c = 0.05, v_adv/c = 0.16 and a (more extreme) case v_adv/c = 0.49. The butterfly indicates the range of values needed to reproduce a neutrino spectrum as observed by IceCube for NGC 1068. The energy density spectrum is expressed in units of the background plasma pressure (see Sect. 2.1). In these units, each individual spectrum has been normalized to an integrated fraction of unity for the sake of illustration.