  The broad-band spectral distributions of non-thermal sources, such as those
of several known blazars, are well described by a log-parabolic fit. The second
degree term in these fits measures the curvature in the spectrum. In this paper
we investigate whether the curvature parameter observed in the spectra of the
synchrotron emission can be used as a fingerprint of stochastic acceleration.
As a first approach we use the multiplicative Central Limit theorem to show how
fluctuations in the energy gain result in the broadening of the spectral shape,
introducing a curvature into the energy distribution. Then, by means of a
Monte-Carlo description, we investigate how the curvature produced in the
electron distribution is linked to the diffusion in momentum space. To get a
more generic description we turn to the diffusion equation in momentum space.
We first study some "standard" scenarios, in order to understand the conditions
that make the curvature in the spectra significant, and the relevance of the
cooling over the acceleration process. We try to quantify the correlation
between the curvature and the diffusive process in the pre-equilibrium stage,
and investigate how the transition between the Klein-Nishina and the Thompson
regime, in Inverse Compton cooling, can determine the curvature in the
distribution at the equilibrium. We apply these results to some observed
trends, such as the anticorrelation between the peak energy and the curvature
term observed in the spectra of Mrk 421, and a sample of BL Lac objects whose
synchrotron emission peaks at X-ray energies.
