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

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