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.