  FRB 121102 is the first fast radio burst source to be spatially associated
with a persistent radio source (QRS121102), the nature of which remains
unknown. We present a detailed observational study of QRS121102 and its host
galaxy. We constrain the physical size of QRS121102 by measuring its
flux-density variability with the VLA in the Ku-band (12 to 18 GHz) and the
K-band (18 to 26 GHz). Any such variability would likely be due to Galactic
refractive scintillation and would require the source radius to be <10^17 cm at
the host-galaxy redshift. We found the radio variability to be lower than the
scintillation theory predictions for such a small source, leaving open the
possibility for non-AGN models for QRS121102. In addition, we roughly estimated
the mass of any potential supermassive black hole (SMBH) associated with
QRS121102 from the width of the H\alpha emission line using a medium-resolution
optical spectrum from the Keck Observatory. The line width gives a velocity
dispersion of <30 km/s, indicating a SMBH mass of <10^{4~5} M_sun. We find the
SMBH mass too low for the observed radio luminosity, and X-ray luminosity
constraints, if QRS121102 were an AGN. Finally, some dwarf galaxies that host
SMBH may be the stripped cores of massive galaxies during the tidal
interactions with companion systems. We find no nearby galaxy at the same
redshift as the QRS121102 host from low-resolution Keck spectra, or from the
PanSTARRS catalog. In conclusion, we find no evidence supporting the hypothesis
that the persistent radio source associated with FRB 121102 is an AGN. We
instead argue that the inferred size, and the flat radio spectrum, favors a
plerion interpretation. We urge continued broadband radio monitoring of
QRS121102 to search for long-term evolution, and the detailed evaluation of
potential analogs that may provide greater insight into the nature of this
class of object.
