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The broad-band spectral distributions of non-thermal sources, such as thoseof several known blazars, are well described by a log-parabolic fit. The seconddegree term in these fits measures the curvature in the spectrum. In this paperwe investigate whether the curvature parameter observed in the spectra of thesynchrotron emission can be used as a fingerprint of stochastic acceleration.As a first approach we use the multiplicative Central Limit theorem to show howfluctuations in the energy gain result in the broadening of the spectral shape,introducing a curvature into the energy distribution. Then, by means of aMonte-Carlo description, we investigate how the curvature produced in theelectron distribution is linked to the diffusion in momentum space. To get amore generic description we turn to the diffusion equation in momentum space.We first study some "standard" scenarios, in order to understand the conditionsthat make the curvature in the spectra significant, and the relevance of thecooling over the acceleration process. We try to quantify the correlationbetween the curvature and the diffusive process in the pre-equilibrium stage,and investigate how the transition between the Klein-Nishina and the Thompsonregime, in Inverse Compton cooling, can determine the curvature in thedistribution at the equilibrium. We apply these results to some observedtrends, such as the anticorrelation between the peak energy and the curvatureterm observed in the spectra of Mrk 421, and a sample of BL Lac objects whosesynchrotron emission peaks at X-ray energies.

Stochastic Acceleration and the Evolution of Spectral Distributions in SSC Sources: A Self Consistent Modeling of Blazars' Flares