  We study the Lindblad master equation in the space of operators and provide
simple criteria for closeness of the hierarchy of equations for correlations.
We separately consider the time evolution of closed and open systems and show
that open systems satisfying the closeness conditions are not necessary of
Gaussian type. In addition, we show that dissipation can induce the closeness
of the hierarchy of correlations in interacting quantum systems. As an example
we study an interacting optomechanical model, the Fermi-Hubbard model, and the
Rabi model, all coupled to a fine-tuned Markovian environment and obtain exact
analytic expressions for the time evolution of two-point correlations.
