  The recently observed X-ray synchrotron emission from four supernova remnants
(SNR) has strengthened the evidence that cosmic ray electrons are accelerated
in SNR. We show, that if this is indeed the case, the local electron spectrum
will be strongly time-dependent, at least above roughly 30 GeV. The time
dependence stems from the Poisson fluctuations in the number of SNR within a
certain volume and within a certain time interval. As far as cosmic ray
electrons are concerned, the Galaxy looks like actively bubbling swiss cheese
rather than a steady, homogeneously filled system. Our finding has important
consequences for studies of the Galactic diffuse gamma-ray emission, for which
a strong excess over model predictions above 1 GeV has been reported recently.
While these models were relying on an electron injection spectrum with index
2.4 -- chosen to fit the local electron flux up to 1 TeV -- we show that an
electron injection index of around 2.0 would a) be consistent with the expected
Poisson fluctuations in the locally observable electron spectrum and b) explain
the above mentioned gamma-ray excess above 1 GeV. An electron injection index
around 2 would also correspond to the average radio synchrotron spectrum of
individual SNR. We use a three-dimensional propagation code to calculate the
spectra of electrons throughout the Galaxy and show that the longitude and
latitude distribution of the leptonic gamma-ray production above 1 GeV is in
accord with the respective distributions for the gamma-ray excess. We finally
point out that our model implies a strong systematic uncertainty in the
determination of the spectrum of the extragalactic gamma-ray background.
