Tobias Grafke
Title: Sharp rare-event asymptotics and fluctuations around instantons
Abstract
Rare and extreme events are notoriously hard to handle in complex stochastic systems: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. Large deviation theory provides a classical way of dealing with events of extremely small probability, but generally only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, and algorithms based upon it, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single computation and without fitting parameters. Notably, prefactor estimates, which take into account fluctuation corrections around the instanton, require the computation of determinants of differential operators. For multiplicative noise, iterated stochastic integrals introduce a second Wiener chaos contribution, so that the resulting prefactor involves a Carleman-Fredholm determinant. Throughout, I will demonstrate the applicability of these methods to high-dimensional real-world systems, for example coming from atmosphere and ocean dynamics.