Gravity is a long-range interaction. As a result, stellar systems are generically (i) inhomogeneous (stars follow intricate orbits), (ii) self-gravitating (stars self-consistenly define the gravitational potential), (iii) resonant (orbits introduce orbital frequencies). In the limit of small perturbations, the efficiency with which a stellar system responds to stimuli is described by linear response theory, a successful framework to predict (linear) modes. Yet, as a stellar systems nears an instability, amplification gets so large that fluctuations are no longer small. This is the realm of non-linear response theory, whose analytical description is much more challenging. This internship focuses on exploring non-linear response theory in stellar systems. For that purpose, we will investigate the "periodic cube", an enlightening self-gravitating toy model. We will explore the dependence of the level of thermal fluctuations, as one lowers the system's dynamical temperature. We will use stochastic methods from renormalisation theory, originating from plasma physics, to predict the associated levels of perturbations. Ultimately, this program of research will offer new clues on non-linear self-gravitating processes, such as mode saturation and statistical correlation functions.