  In this article, we use quasifree reduction to derive the time-dependent
Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum
fluctuations around a Bose-Einstein condensate in $\mathbb R^d$. We prove
global well-posedness for the HFB equations for sufficiently regular pair
interaction potentials, and establish key conservation laws. Moreover, we show
that the solutions to the HFB equations exhibit a symplectic structure, and
have a form reminiscent of a Hamiltonian system. In particular, this is used to
relate the HFB equations to the HFB eigenvalue equations encountered in the
physics literature. Furthermore, we construct the Gibbs states at positive
temperature associated with the HFB equations, and establish criteria for the
emergence of Bose-Einstein condensation.
