Here we investigate some aspects of stochastic acceleration of ultrarelativistic electrons by magnetic turbulence. In particular, we discuss the steady-state energy spectra of particles undergoing momentum diffusion due to resonant interactions with turbulent MHD modes, taking rigorously into account direct energy losses connected with different radiative cooling processes. For the magnetic turbulence we assume a given power spectrum of the type W(k) ∝ k−q. In contrast to the previous approaches, however, we assume a finite range of turbulent wavevectors k, consider a variety of turbulence spectral indexes 1 ≤ q ≤ 2, and concentrate on the case of a very inefficient particle escape from the acceleration site. We find that for different cooling and injection conditions, stochastic acceleration processes tend to establish a modified ultrarelativistic Maxwellian distribution of radiating particles, with the high-energy exponential cutoff shaped by the interplay between cooling and acceleration rates. For example, if the timescale for the dominant radiative process scales with the electron momentum as ∝ pr, the resulting electron energy distribution is of the form ne(p) ∝ p2 exp [ −1 a (p/peq) a]
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