A stochastic quasi-classical wavefunction of the Universe from the third
  quantization procedure
  (abbreviated) We study quantized solutions of WdW equation describing a
closed FRW universe with a $\Lambda $ term and a set of massless scalar fields.
We show that when $\Lambda \ll 1$ in the natural units and the standard
$in$-vacuum state is considered, either wavefunction of the universe, $\Psi$,
or its derivative with respect to the scale factor, $a$, behave as random
quasi-classical fields at sufficiently large values of $a$, when $1 \ll a \ll
e^{{2\over 3\Lambda}}$ or $a \gg e^{{2\over 3\Lambda}}$, respectively.
Statistical r.m.s value of the wavefunction is proportional to the
Hartle-Hawking wavefunction for a closed universe with a $\Lambda $ term.
Alternatively, the behaviour of our system at large values of $a$ can be
described in terms of a density matrix corresponding to a mixed state, which is
directly determined by statistical properties of $\Psi$. It gives a non-trivial
probability distribution over field velocities. We suppose that a similar
behaviour of $\Psi$ can be found in all models exhibiting copious production of
excitations with respect to $out$-vacuum state associated with classical
trajectories at large values of $a$. Thus, the third quantization procedure may
provide a 'boundary condition' for classical solutions of WdW equation.
