  We investigate a case of the Hu-Paz-Zhang master equation of the
Caldeira-Leggett model without Lindblad form obtained in the weak-coupling
limit up to the second-order perturbation. In our study, we use Gaussian
initial states to be able to employ a sufficient and necessary condition, which
can expose positivity violations of the density operator during the time
evolution. We demonstrate that the evolution of the non-Markovian master
equation has problems when the stationary solution is not a positive operator,
i.e., does not have physical interpretation. We also show that solutions always
remain physical for small-times of evolution. Moreover, we identify a strong
anomalous behavior, when the trace of the solution is diverging. We also
provide results for the corresponding Markovian master equation and show that
positivity violations occur for various types of initial conditions even when
the stationary solution is a positive operator. Based on our numerical results,
we conclude that this non-Markovian master equation is superior to the
corresponding Markovian one.
