  Starting from the forward and backward infinitesimal generators of bilateral,
time-homogeneous Markov processes, the self-adjoint Hamiltonians of the
generalized Schroedinger equations are first introduced by means of suitable
Doob transformations. Then, by broadening with the aid of the Dirichlet forms
the results of the Nelson stochastic mechanics, we prove that it is possible to
associate bilateral, and time-homogeneous Markov processes to the wave
functions stationary solutions of our generalized Schroedinger equations.
Particular attention is then paid to the special case of the Levy-Schroedinger
equations and to their associated Levy-type Markov processes, and to a few
examples of Cauchy background noise.
