  In the paper, we discuss the studies of mathematical models of diffusion
scattering of waves in the phase space, and relation of these models with
quantum mechanics. In the previous works it is shown that in these models of
classical scattering process of waves, the quantum mechanical description
arises as the asymptotics after a small time. In this respect, the proposed
models can be considered as examples in which the quantum descriptions arise as
approximate ones for certain hypothetical reality. The deviation between the
proposed models and the quantum ones can arise, for example, for processes with
rapidly changing potential function. Under its action the diffusion scattering
process of waves will go out from the states described by quantum mechanics.
  In the paper it is shown that the proposed models of diffusion scattering of
waves possess the property of gauge invariance. This implies that they are
described similarly in all inertial coordinate systems, i.e., they are
invariant under the Galileo transformations.
  We propose a program of further research.
