A kinetic theory of diffusion in general relativity with cosmological
  scalar field
  A new model to describe the dynamics of particles undergoing diffusion in
general relativity is proposed. The evolution of the particle system is
described by a Fokker-Planck equation without friction on the tangent bundle of
spacetime. It is shown that the energy-momentum tensor for this matter model is
not divergence-free, which makes it inconsistent to couple the Fokker-Planck
equation to the Einstein equations. This problem can be solved by postulating
the existence of additional matter fields in spacetime or by modifying the
Einstein equations. The case of a cosmological scalar field term added to the
left hand side of the Einstein equations is studied in some details. For the
simplest cosmological model, namely the flat Robertson-Walker spacetime, it is
shown that, depending on the initial value of the cosmological scalar field,
which can be identified with the present observed value of the cosmological
constant, either unlimited expansion or the formation of a singularity in
finite time will occur in the future. Future collapse into a singularity also
takes place for a suitable small but positive present value of the cosmological
constant, in contrast to the standard diffusion-free scenario.
