  Solar-like oscillations in red giants have been investigated with CoRoT and
Kepler, while pulsations in more evolved M giants have been studied with
ground-based microlensing surveys. After 3.1 years of observation with Kepler,
it is now possible to make a link between these different observations of
semi-regular variables. We aim to identify period-luminosity sequences in
evolved red giants identified as semi-regular variables. Then, we investigate
the consequences of the comparison of ground-based and space-borne
observations. We have first measured global oscillation parameters of evolved
red giants observed with Kepler with the envelope autocorrelation function
method. We then used an extended form of the universal red giant oscillation
pattern, extrapolated to very low frequency, to fully identify their
oscillations. From the link between red giant oscillations observed by Kepler
and period-luminosity sequences, we have identified these relations in evolved
red giants as radial and non-radial solar-like oscillations. We were able to
expand scaling relations at very low frequency. This helped us to identify the
different sequences of period-luminosity relations, and allowed us to propose a
calibration of the K magnitude with the observed frequency large separation.
Interpreting period-luminosity relations in red giants in terms of solar-like
oscillations allows us to investigate, with a firm physical basis, the time
series obtained from ground-based microlensing surveys. This can be done with
an analytical expression that describes the low-frequency oscillation spectra.
The different behavior of oscillations at low frequency, with frequency
separations scaling only approximately with the square root of the mean stellar
density, can be used to address precisely the physics of the semi-regular
variables.
