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

Abstract

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.

Symmetries and Zero Modes in Sample Path Large Deviations

“…As an additional, concrete motivation to go beyond such rough estimates in practical applications, it has been pointed out very recently that for assessing the relative importance of different instantonic transition paths, knowledge of the LDT prefactor at leading order may be vital even at comparably small noise strengths [29]. In the last year, there has been a lot of activity to provide generic numerical tools that also allow for the computation of the leading order term of the large deviation prefactor for the statistics of final time observables of small noise ordinary SDEs using symmetric Riccati matrix differential equations, either forward or backward in time [30][31][32][33]. In an abstract setting, expressions for prefactors in this context, even at arbitrarily high order, have already been known rigorously since the 1980's [2,[34][35][36] and are, not surprisingly, related to a certain operator determinant at the leading order.…”

“…This is advantageous if either, from a numerical point of view, the spatial dimension of the system is not too large, with the Riccati matrix being of size n × n for a n-dimensional SDE, or if an analytical analysis of the resulting equations is desired. Our first contribution in this paper is to make the connection to functional determinants more precise and to add to the existing derivations of the Riccati equations using (i) a WKB analysis of the Kolmogorov backward equation [31] (ii) a discretization approach of the path integral [30] or (iii) the use of the Feynman-Kac formula for Gaussian fluctuations [30,33] a fourth derivation that makes explicit use of Forman's theorem. Furthermore, in contrast to previous derivations, we also include the case of Itô SDEs with multiplicative noise here.…”

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

“…where here X ,s,y is a single particle satisfying, e.g., the small noise SDE: dX ,s,y t = b(X ,s,y t )dt + √ σ(X ,s,y t )dW t (5) X ,s,y s = y ∈ R d have been generalized in order to design importance sampling schemes for problems related to metastable dynamics of the stochastic system such as exit probabilities and mean first passage times -see e.g. [55] and the references therein. Thus we view this paper as the first step towards a new method of designing importance sampling schemes for studying properties of the empirical measure which are of interest to the greater scientific community, such as the free-energy differences for chemical and biological molecular systems [3,28,59,71,76] or exit times of the empirical measure from domains of attraction in models related to social dynamics and consensus convergence [6,31,47,48,57].…”

“…there is a one-to-one mapping between R dN /S N and P N (R d ) (see e.g. [27] Equation (55) or [56] Equation (1)). Hence it is natural to assume the functions G N from Equation (3) are such that we can find G :…”

Diffusivity dependence of the transition path ensemble

“…Rather, it is vital that fluctuations around this path be incorporated. We believe that both this fundamental counter-intuitive insight, as well as our MCMC method, will be of value for going beyond the regime of asymptotically low diffusivity in both large deviations theory [18,47,48] and the study of rare events [49] as well as transition path ensembles [23][24][25][26].…”