  We suggest that the energy source of the observed diffuse gamma-ray emission
from the direction of the Galactic center is the Galactic black hole Sgr A*,
which becomes active when a star is captured at a rate of $\sim 10^{-5} $
yr^{-1}. Subsequently the star is tidally disrupted and its matter is accreted
into the black hole. During the active phase relativistic protons with a
characteristic energy $\sim 6\times 10^{52}$ erg per capture are ejected. Over
90% of these relativistic protons disappear due to proton-proton collisions on
a timescale $\tau_{pp} \sim 10^4$ years in the small central bulge region with
radius $\sim 50$ pc within Sgr A*, where the density is $\ge 10^3$ cm^{-3}. The
gamma-ray intensity, which results from the decay of neutral pions produced by
proton-proton collisions, decreases according to $e^{-t/\tau_{pp}}$, where t is
the time after last stellar capture. Less than 5% of relativistic protons
escaped from the central bulge region can survive and maintain their energy for
>10^7 years due to much lower gas density outside, where the gas density can
drop to $\sim 1$ cm$^{-3}$. They can diffuse to a $\sim 500$ pc region before
disappearing due to proton-proton collisions. The observed diffuse GeV
gamma-rays resulting from the decay of neutral pions produced via collision
between these escaped protons and the gas in this region is expected to be
insensitive to time in the multi-injection model with the characteristic
injection rate of 10^{-5} yr^{-1}. Our model calculated GeV and 511 keV
gamma-ray intensities are consistent with the observed results of EGRET and
INTEGRAL, however, our calculated inflight annihilation rate cannot produce
sufficient intensity to explain the COMPTEL data.
