  Statistically interpretable axioms are formulated that define a quantum
stochastic process (QSP) as a causally ordered operator field in an arbitrary
space-time region T of an open quantum system under a sequential observation at
a discrete space-time localization. It is shown that to every QSP described in
the weak sense by a self-consistent system of causally ordered correlation
kernels there corresponds a unique, up to unitary equivalence, minimal QSP in
the strong sense. It is shown that the proposed QSP construction, which reduces
in the case of the linearly ordered discrete T=Z to the construction of the
inductive limit of Lindblad's canonical representations, corresponds to
Kolmogorov's classical reconstruction if the order on T is ignored and leads to
Lewis construction if one uses the system of all (not only causal) correlation
kernels, regarding this system as lexicographically preordered on T. The
approach presented encompasses both nonrelativistic and relativistic
irreversible dynamics of open quantum systems and fields satisfying the
conditions of local commutativity and semigroup covariance. Also given are
necessary and sufficient conditions of dynamicity (or conditional Markovianity)
and regularity, these leading to the properties of complete mixing (relaxation)
and ergodicity of the QSP.
