Sampling rare trajectories in stochastic systems

Sara Oliver Bonafoux (Supervisors: Raúl Toral, Tobias Galla)
Master Thesis (2022)

Despite their infrequent occurrence, rare events in stochastic processes can lead to the most catastrophic outcomes. Much interest has recently been focused on the sampling of rare tra- jectories and the quantification of their statistics in models of stochastic phenomena. This problem is computationally demanding if conventional sampling methods are used, so specific rare-trajectory sampling techniques must be developed to deal with it. The renowned Wentzel- Kramers-Brillouin (WKB) method constitutes a tool to characterise most likely paths describing rare events in the limit of small noise, but is incapable of describing the statistics of rare trajec- tories in systems with finite stochasticity. A recently proposed backtracking sampling method that overcomes this limitation consists of working with so-called stochastic bridges, which are trajectories generated backwards in time that are constrained to have fixed start and end points.
In this project we explore the WKB and backtracking formalisms in order to sample rare trajectories in three stochastic models. We first focus on the fading of an epidemic in the SIS model, reproducing existing results in the literature. We next study the extinction of a population in a model of chemical reactions for which no rare trajectories have previously been generated, this being a new contribution of our work. Finally, we focus on the escape of a Brownian particle from a double-well potential, proposing the backtracking method as a simpler alternative to the techniques used in the literature to generate trajectories of this rare stochastic phenomenon. The application of the backtracking method to this model is also a new contribution of the thesis. In all cases we find that at finite noise levels the stochastic bridges capture fluctuations around the WKB optimal path. In addition, we show that the WKB formalism in incapable of characterising most likely paths connecting two stable states, while the backtracking method can be successfully used to sample trajectories that transit from one attracting state to another.

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