  A new exactly solvable model for the evolution of relativistic kinetic system
interacting with an internal stochastic reservoir under the influence of a
gravitational background expansion is established. This model of
self-interaction is based on the relativistic kinetic equation for the
distribution function defined in the extended phase space. The supplementary
degree of freedom is described by the scalar stochastic variable (Langevin
source), which is considered to be the constructive element of the effective
one-particle force. The expansion of the Universe is shown to be accelerated
for the suitable choice of the non-minimal self-interaction force.
