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We obtain the renormalized equations of motion for matter and semi-classicalgravity in an inhomogeneous space-time. We use the functional Schrodingerpicture and a simple Gaussian approximation to analyze the time evolution ofthe $\lambda\phi^4$ model, and we establish the renormalizability of thisnon-perturbative approximation. We also show that the energy-momentum tensor inthis approximation is finite once we consider the usual mass and couplingconstant renormalizations, without the need of further geometricalcounter-terms.Comment: 22 page

QUANTUM EVOLUTION OF INHOMOGENEITIES IN CURVED SPACE