  In this article we define and investigate statistical operators and an
entropy functional for Bernstein stochastic processes associated with
hierarchies of forward-backward systems of decoupled deterministic linear
parabolic partial differential equations. The systems under consideration are
defined on open bounded domains $D\subset \mathbb{R}^{d}$ of Euclidean space
where $d\in \mathbb{N}^{+}$ is arbitrary, and are subject to Neumann boundary
conditions. We assume that the elliptic part of the parabolic operator in the
equations is a self-adjoint Schr\"odinger operator,bounded from below and with
compact resolvent in $L^{2}(D)$. The statistical operators we consider are then
trace-class operators defined from sequences of probabilities associated with
the point spectrum of the elliptic part in question, which allow the
distinction between pure and mixed processes. We prove in particular that the
Bernstein processes of maximal entropy are those for which the associated
sequences of probabilities are of Gibbs type. We illustrate our results by
considering processes associated with a specific hierarchy of forward-backward
heat equations defined in a two-dimensional disk.
