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

Sharp Asymptotic Estimates for Expectations, Probabilities, and Mean First Passage Times in Stochastic Systems with Small Noise