Metastability for a PDE with blow-up and the FFG dynamics on diluted models

This thesis consists of two separate parts: in each we study the stability under small perturbations of certain probability models in different contexts. In the first, we study small random perturbations of a deterministic dynamical system and show that these are unstable, in the sense that the perturbed systems have a different qualitative behavior than that of the original system. In the second part we situate ourselves in the context of statistical mechanics, where we study the stability of equilibrium infinite-volume measures under small deterministic perturbations in the parameters of the model. More precisely, we show that Gibbs measures for a general class of systems are continuous with respect to changes in the interaction and/or density of particles and, hence, stable under small perturbations of them. We also study under which conditions do certain typical configurations of these systems remain stable in the zero-temperature limit.