  The work analyzes the stability of the quantum eigenstates when they are
submitted to fluctuations by using the stochastic generalization of the
Madelung quantum hydrodynamic approach. In the limit of sufficiently slow
kinetics, the quantum eigenstates show to remain stationary configurations with
a very small perturbation of their mass density distribution. The work shows
that the stochastic quantum hydrodynamic model allows to obtain the definition
of the quantum eigenstates without recurring to the measurement process or any
reference to the classical mechanics, by identifying them from their intrinsic
properties of stationarity and stability. By using the discrete approach, the
path integral solution of the stochastic quantum-hydrodynamic equation has been
derived in order to investigate how the final stationary configurations depend
by the the initial condition of the quatum superposition of states. The
stochastic quantum hydrodynamics shows that the superposition of states can
relax to different stationary states that, in the small noise limit, are the
slightly perturbed quantum eigenstates. The work shows that the final
stationary eigenstate depends by the initial configuration of the superposition
of states and that possibly the probability transition to each eigenstates can
satisfy the Born rule, allowing the decoherence process to be compatible with
the Copenhagen interpretation of quantum mechanics.
