  Context: Since a few years, signatures of supernova remnants associated with
molecular clouds have been detected in gamma rays. Whether these gamma rays are
generated by cosmic ray electrons or by cosmic ray protons is usually not
known. The detection of hadronic ionization signatures in spatial coincidence
with gamma ray signatures can help to unambiguously identify supernova remnants
as sources of cosmic ray protons.
  Methods: In order to calculate hadronic signatures from cosmic ray-induced
ionization for an examination of the origin of the observed gamma rays, the
transport equation for cosmic ray protons propagating in a molecular cloud,
including the relevant momentum loss processes, is solved analytically and the
proton flux at any position in the cloud is determined.
  Results: Since the solution of the transport equation is obtained for
arbitrary source functions, it can be used for a variety of supernova remnants.
The corresponding theoretical ionization rate, as a function of the penetration
depth, is derived and compared to photoinduced ionization profiles in a case
study with four supernova remnants associated with molecular clouds. Three of
the remnants show a clear dominance of the hadronically induced ionization
rate, while for one remnant, X-ray emission seems to dominate by a factor of
10.
  Conclusions: This is the first derivation of position-dependent profiles for
cosmic ray-induced ionization with an analytic solution for arbitrary cosmic
ray source spectra. The cosmic ray-induced ionization has to be compared to
X-ray ionization for strong X-ray sources. For sources dominated by cosmic
ray-induced ionization (e.g., W49B), the ionization profiles can be used in the
future to map the spatial structure of hadronic gamma rays and
rotation-vibrational lines induced by cosmic ray protons, helping to identify
sources of hadronic cosmic rays.
