  In this study we present an analysis of the performance and properties of the
quasi-periodic (QP) GP kernel, which is the multiplication of the
squared-exponential kernel by the exponential-sine-squared kernel, based on an
extensive set of synthetic RVs, into which the signature of activity was
injected. We find that while the QP-GP rotation parameter matches the simulated
rotation period of the star, the length scale cannot be directly connected to
the spot lifetimes on the stellar surface. Regarding the setup of the priors
for the QP-GP, we find that it can be advantageous to constrain the QP-GP
hyperparameters in different ways depending on the application and the goal of
the analysis. We find that a constraint on the length scale of the QP-GP can
lead to a significant improvement in identifying the correct rotation period of
the star, while a constraint on the rotation hyperparameter tends to lead to
improved planet detection efficiency and more accurately derived planet
parameters. Even though for most of the simulations the Bayesian evidence
performed as expected, we identified not far-fetched cases where a blind
adoption of this metric would lead to wrong conclusions. We conclude that
modeling stellar astrophysical noise by using a QP-GP considerably improves
detection efficiencies and leads to precise planet parameters. Nevertheless,
there are also cases in which the QP-GP does not perform optimally, for example
RV variations dynamically evolving on short timescales or a mixture of a very
stable activity component and random variations. Knowledge of these limitations
is essential for drawing correct conclusions from observational data.
