  We analyse processes of electron acceleration in the Fermi Bubbles in order
to define parameters and restrictions of the models, which are suggested for
the origin of these giant radio and gamma-ray structures. In the case of
leptonic origin of the nonthermal radiation from the Bubbles, these electrons
should be produced somehow in-situ because of relatively short lifetime of high
energy electrons, which lose their energy by synchrotron and inverse Compton
processes. It has been suggested that electrons in Bubbles may be accelerated
by shocks produced by tidal disruption of star accreting onto the central black
hole or a process of re-acceleration of electrons ejected by supernova
remnants. These processes will be investigated in subsequent papers. In this
paper we focus to study in-situ stochastic (Fermi) acceleration by a
hydromagnetic/supersonic turbulence, in which electrons can be directly
accelerated from the background plasma. We showed that the acceleration from
the background plasma is able to explain the observed fluxes of radio and
gamma-ray emission from the Bubbles but the range of permitted parameters of
the model is strongly restricted.
