Generation of stochastic trajectories: applications to complex systems

The generation of stochastic trajectories is one of the preferred tools to study complex systems. To begin with, the analysis of trajectories gives important information at a simple glimpse (e.g. relevant scales, size of fluctuations, the duration of transient regimes, or the presence of stationary states). Furthermore, the statistical characterization of trajectories unveils quantitative descriptions for the processes, since it allows computing averages and probability distributions for observables of interest. This thesis first focuses on reviewing the principal methods to generate stochastic trajectories, comparing their strengths and key features. Then, these methods are used to address two questions in the context of epidemiology: What is the effect of the mobility structure of cities in the spreading of infectious diseases? And what is the mechanism behind the long epidemic survival times widely observed in COVID- 19 data? Finally, we will enter the problem of the generation of rare trajectories, which are inaccessible for standard algorithms in feasible times. We present a new method to sample such rare trajectories coined the “backtracking method”. The inspection of stochastic trajectories will be recurrently complemented with standard techniques from the theory of stochastic process, which are also presented in a self- contained manner, such as the computation of absorption times with the backward Kolmogorov equations or the Wentzel-Kramers-Brillouin method.