  We describe space--time fluctuations by means of small fluctuations of the
metric on a given background metric. From a minimally coupled Klein--Gordon
equation we obtain within a weak-field approximation up to second order and an
averaging procedure over a finite space--time scale given by the quantum
particle in the non--relativistic limit a modified Schr\"odinger equation. The
dominant modification consists in an anomalous inertial mass tensor which
depends on the type of particle and on the fluctuation scenario. The scenario
considered in this paper is a most simple picture of spacetime fluctuations and
gives an existence proof for an apparent violation of the weak equivalence
principle and, in general, for a violation of Lorentz invariance.
