Learning control has become an appealing alternative to the derivation of control laws based on classic control theory. However, a major shortcoming of learning control is the lack of performance guarantees which prevents its application in many real-world scenarios. As a step in this direction, we provide a stability analysis tool for controllers acting on dynamics represented by Gaussian processes (GPs). We consider arbitrary Markovian control policies and system dynamics given as (i) the mean of a GP, and (ii) the full GP distribution. For the first case, our tool finds a state space region, where the closed-loop system is provably stable. In the second case, it is well known that infinite horizon stability guarantees cannot exist. Instead, our tool analyzes finite time stability. Empirical evaluations on simulated benchmark problems support our theoretical results.