Quantum propagators in a random metric

We consider second-order differential operators with coefficients which are Gaussian random fields. When the covariance becomes singular at short dist ances then the propagators of the Schrodinger equation as well as of the wa ve equation behave in an anomalous way. In particular, the Feynman propagat or for the wave equation is less singular than the one with deterministic c oefficients. We suggest some applications to quantum gravity.