We analyse the influence of the stochastic particle acceleration for the
evolution of the electron spectrum. We assume that all investigated spectra are
generated inside a spherical, homogeneous source and also analyse the
synchrotron and inverse Compton emission generated by such an object. The
stochastic acceleration is treated as the diffusion of the particle momentum
and is described by the momentum-diffusion equation. We investigate the
stationary and time dependent solutions of the equation for several different
evolutionary scenarios. The scenarios are divided into two general classes.
First, we analyse a few cases without injection or escape of the particles
during the evolution. Then we investigate the scenarios where we assume
continuous injection and simultaneous escape of the particles. In the case of
no injection and escape the acceleration process, competing with the radiative
cooling, only modifies the initial particle spectrum. The competition leads to
a thermal or quasi-thermal distribution of the particle energy. In the case of
the injection and simultaneous escape the resulting spectra depend mostly on
the energy distribution of the injected particles. In the simplest case, where
the particles are injected at the lowest possible energies, the competition
between the acceleration and the escape forms a power-law energy distribution.
We apply our modeling to the high energy activity of the blazar Mrk 501
observed in April 1997. Calculating the evolution of the electron spectrum
self-consistently we can reproduce the observed spectra well with a number of
free parameters that is comparable to or less than in the "classic stationary"
one--zone synchrotron self-Compton scenario.Comment: 11 pages, 4 figures, accepted for publication in A&