Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise

Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimizer of the Freidlin-Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction-advection-diffusion type.