  We study the statistical properties of an ensemble of weak gravitational
waves interacting nonlinearly in a flat space-time. We show that the resonant
three-wave interactions are absent and develop a theory for four-wave
interactions in a reduced case of a diagonal metric tensor. In this limit,
where only one type of gravitational waves are present, we derive the
interaction Hamiltonian and consider the asymptotic regime of weak
gravitational wave turbulence. Both direct and inverse cascades are found for
the energy and the wave action respectively, and the corresponding wave spectra
are derived. The inverse cascade is characterized by a finite-time propagation
of the metric excitations - a process similar to an explosive non-equilibrium
Bose-Einstein condensation, which provides an efficient mechanism to ironing
out small-scale inhomogeneities. The direct cascade leads to an accumulation of
the radiation energy in the system. These processes might be important for
understanding the early Universe where a background of weak nonlinear
gravitational waves is expected.
