Consider the behaviour of a deterministic dynamical system in a domain containing a stable attracting equilibrium — for example, a particle trapped in a potential well, or a marble dropped into a large bowl. Given suitable initial conditions, the system will move towards the equilibrium and stay there once it has arrived at the equilibrium. However, if the dynamical system is perturbed by white noise, is it possible that the system might exit the domain of attraction? If it is possible, what conditions do we need? Where and when is the system most likely to exit the domain?
These are the questions involved in the “exit problem”, which is often encountered in large deviations theory. In this talk we will take a different perspective from large deviations theory, and instead apply ideas from control theory to answer these questions in the context of linear systems.