Self-Similarity and Large Deviations in Two Subordinated Stochastic Processes

Large deviation theory is the toolbox of theoretical results in probability theory which underlies most of equilibrium and nonequilibrium statistical mechanics. The central phenomenon in large deviation theory is the concentration of the probability for a properly scaled stochastic sequence around its mean. This is an effect of the asymptotic dominance of the second cumulant of the scaled stochastic sequence with respect to higher-order cumulants. A fundamental result in statistical mechanics concerns the behavior of macroscopic systems around critical points (second order or continuous phase transitions), which is understood in terms of anomalous scaling properties. However, to sustain the scaling around an anomalous probability distribution, all cumulants must asymptotically contribute. There is thus an open question regarding the standpoint of large deviation theory with respect to critical phenomena. This thesis aims at addressing this question in the context of stochastic processes. After an overview of the Langevin and Fokker-Planck descriptions for Brownian motion, the anomalous behavior will be introduced in terms of a subordination mechanism, which is employed for instance in continuous-time random walks, in diffusing-diffusivities models, and in the description of the motion of polymers in the grand-canonical ensemble. The thesis will involve analytical insight and possibly numerical methods.