On a heuristic point of view concerning the motion of matter. From   random metric to Schroedinger equation
  The motion of a particle is studied in a random space-time. It is assumed that the velocity is small enough for the non-relativistic approximation to be valid. The randomness of the metric induces a diffusion in coordinate space. Hence it is shown that the evolution of the probability density is given by Schroedinger equation. 
Emilio Santos