  In this article we investigate the properties of Bernstein processes
generated by infinite hierarchies of forward-backward systems of decoupled
linear deterministic parabolic partial differential equations defined in Rd,
where d is arbitrary. An important feature of those systems is that the
elliptic part of the parabolic operators may be realized as an unbounded
Schr\"odinger operator with compact resolvent in standard L2-space. The
Bernstein processes we are interested in are in general non-Markovian, may be
stationary or non-stationary and are generated by weighted averages of measures
naturally associated with the pure point spectrum of the operator. We also
introduce time-dependent trace-class operators which possess most of the
attributes of density operators in Quantum Statistical Mechanics, and prove
that the statistical averages of certain bounded self-adjoint observables
usually evaluated by means of such operators coincide with the expectation
values of suitable functions of the underlying processes. In the particular
case where the given parabolic equations involve the Hamiltonian of an
isotropic system of quantum harmonic oscillators, we show that one of the
associated processes is identical in law with the periodic Ornstein-Uhlenbeck
process.
