  We discuss Einstein gravity for a fluid consisting of particles interacting
with an unidentified environment of some other particles whose dissipative
effect is approximated by a diffusion. The environment is described by a time
dependent cosmological term which is compensating the lack of the conservation
law of the energy momentum of the diffusing fluid. We are interested in a
homogeneous flat expanding Universe described by a scale factor $a$. For a
fluid of massless particles at finite temperature we obtain explicit solutions
of the diffusion equation which are in the form of a modified Juttner
distribution with a time dependent temperature. At later time Universe
evolution is described as a diffusion at zero temperature with no
equilibration. We find solutions of the diffusion at zero temperature which can
be treated as a continuation to a later time of the finite temperature
solutions describing an early stage of the Universe. A conservation of the
total energy momentum determines the cosmological term up to a constant. The
resulting energy momentum inserted into Einstein equations gives a modified
Friedmann equation. Solutions of the Friedmann equation depend on the initial
value of the cosmological term. The large value of the cosmological constant
implies an exponential expansion. If the initial conditions allow a power-like
solution for a large time then it must be of the form $a\simeq \tau$ (no
deceleration, $\tau$ is the cosmic time). The final stage of the Universe
evolution is described by a non-relativistic diffusion of a cold dust.
