  We develop a general framework for effective equations of expectation values
in quantum cosmology and pose for them the quantum Cauchy problem with
no-boundary and tunneling wavefunctions. Cosmological configuration space is
decomposed into two sectors that give qualitatively different contributions to
the radiation currents in effective equations. The field-theoretical sector of
inhomogeneous modes is treated by the method of Euclidean effective action,
while the quantum mechanical sector of the spatially homogeneous inflaton is
handled by the technique of manifest quantum reduction to gauge invariant
cosmological perturbations. We apply this framework in the model with a big
negative non-minimal coupling, which incorporates a recently proposed low
energy (GUT scale) mechanism of the quantum origin of the inflationary Universe
and study the effects of the quantum inflaton mode.
