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(abbreviated) We study quantized solutions of WdW equation describing aclosed 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 randomquasi-classical fields at sufficiently large values of $a$, when $1 \ll a \lle^{{2\over 3\Lambda}}$ or $a \gg e^{{2\over 3\Lambda}}$, respectively.Statistical r.m.s value of the wavefunction is proportional to theHartle-Hawking wavefunction for a closed universe with a $\Lambda $ term.Alternatively, the behaviour of our system at large values of $a$ can bedescribed in terms of a density matrix corresponding to a mixed state, which isdirectly determined by statistical properties of $\Psi$. It gives a non-trivialprobability distribution over field velocities. We suppose that a similarbehaviour of $\Psi$ can be found in all models exhibiting copious production ofexcitations with respect to $out$-vacuum state associated with classicaltrajectories at large values of $a$. Thus, the third quantization procedure mayprovide a 'boundary condition' for classical solutions of WdW equation.

A stochastic quasi-classical wavefunction of the Universe from the third quantization procedure