Spatial distribution of exoplanet candidates based on Kepler and $Gaia$
  data
  The spatial distribution of exoplanets in the Galaxy is important for our
understanding of planet formation and evolution. We aim to determine the
spatial gradients of exoplanet occurrence in the Solar neighbourhood and in the
vicinity of open clusters. We combined Kepler and Gaia DR2 data for this
purpose, splitting the volume sampled by the Kepler mission into certain
spatial bins. We determined an uncorrected and bias-corrected exoplanet
frequency and metallicity for each bin. There is a clear drop in the
uncorrected exoplanet frequency with distance for F-type stars, a decline with
increasing distance along the Galactic longitude l=90 deg, and a drop with
height above the Galactic plane. We find that the metallicity behaviour cannot
be the reason for the drop of the exoplanet frequency around F stars with
increasing distance. We argue that the above-mentioned gradients of uncorrected
exoplanet frequency result from a single bias of undetected smaller planets
around fainter stars. When we correct for observational biases, most of these
gradients in exoplanet frequency become statistically insignificant. Only a
slight decline of the planet occurrence with distance for F stars remains
significant at the 3 sigma level. Apart from that, the spatial distribution of
exoplanets in the Kepler field of view is compatible with a homogeneous one. At
the same time, we do not find a significant change in the exoplanet frequency
with increasing distance from open clusters. As a byproduct, we identified six
exoplanet host star candidates that are members of open clusters. Four of them
are in the NGC 6811 (KIC 9655005, KIC 9533489, Kepler-66, Kepler-67) and two
belong to NGC 6866 (KIC 8396288, KIC 8331612). Two out of the six had already
been known to be cluster members.
