  This paper considers open quantum systems whose dynamic variables satisfy
canonical commutation relations and are governed by Markovian
Hudson-Parthasarathy quantum stochastic differential equations driven by
external bosonic fields. The dependence of the Hamiltonian and the system-field
coupling operators on the system variables is represented using the Weyl
functional calculus. This leads to an integro-differential equation (IDE) for
the evolution of the quasi-characteristic function (QCF) which encodes the
dynamics of mixed moments of the system variables. Unlike quantum master
equations for reduced density operators, this IDE involves only complex-valued
functions on finite-dimensional Euclidean spaces and extends the Wigner-Moyal
phase-space approach for quantum stochastic systems. The dynamics of the QCF
and the related Wigner quasi-probability density function (QPDF) are discussed
in more detail for the case when the coupling operators depend linearly on the
system variables and the Hamiltonian has a nonquadratic part represented in the
Weyl quantization form. For this class of quantum stochastic systems, we also
consider an approximate computation of invariant states and discuss the
deviation from Gaussian quantum states in terms of the $\chi^2$-divergence (or
the second-order Renyi relative entropy) applied to the QPDF. The results of
the paper may find applications to investigating different aspects of the
moment stability, relaxation dynamics and invariant states in open quantum
systems.
