  Gaussian Process State Space Models (GP-SSM) are a data-driven stochastic
model class suitable to represent nonlinear dynamics. They have become
increasingly popular in non-parametric modeling approaches since they provide
not only a prediction of the system behavior but also an accuracy of the
prediction. For the application of these models, the analysis of fundamental
system properties is required. In this paper, we analyze equilibrium
distributions and stability properties of the GP-SSM. The computation of
equilibrium distributions is based on the numerical solution of a Fredholm
integral equation of the second kind and is suitable for any covariance
function. Besides, we show that the GP-SSM with squared exponential covariance
function is always mean square bounded and there exists a set which is positive
recurrent.
