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

Tobias Grafke, Tobias Schäfer+1 more|arXiv (Cornell University)|Mar 8, 2021

Ecosystem dynamics and resilience4 citations

TL;DR

This paper presents a practical framework for computing sharp asymptotic estimates of expectations, probabilities, and mean first passage times in small-noise stochastic systems by deriving explicit formulas for prefactors using matrix Riccati equations. The method leverages Freidlin-Wentzell large deviation theory and computes next-order corrections via solutions to Riccati equations driven by the instanton path, enabling accurate, numerically tractable estimates even in high- and infinite-dimensional settings such as SPDEs.

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.

Motivation & Objective

• To address the lack of efficient numerical methods for computing prefactor corrections in large deviation theory beyond exponential scaling.
• To provide explicit, numerically implementable formulas for sharp asymptotic estimates of expectations, probabilities, and mean first passage times in small-noise stochastic systems.
• To extend existing instanton-based methods to compute absolute-scale probabilities and expectations, including over the invariant measure and for infinite-time or infinite-dimensional systems.
• To demonstrate the method’s accuracy and efficiency on finite- and infinite-dimensional stochastic PDEs, including irreversible and nonlinear dynamics.

Proposed method

• The method computes prefactor corrections by solving well-posed forward or backward matrix Riccati equations that depend on the Freidlin-Wentzell instanton path.
• The Riccati equations are derived from second-order variations of the action functional and encode the fluctuation determinant around the most likely path.
• For finite-time quantities, the approach uses time-ordered Riccati solutions with initial or final conditions tailored to the target estimate (e.g., expectations, exit probabilities).
• For invariant measure quantities, the method employs Riccati equations with periodic or steady-state boundary conditions to capture long-time statistical behavior.
• The framework is generalized to non-Gaussian noise via path integral methods and applied to jump processes, showing broad applicability.
• Numerical validation is performed using direct sampling (e.g., Gillespie algorithm) and comparison with instanton predictions, confirming accuracy even for small noise.

Experimental results

Research questions

• RQ1 How can prefactor corrections be systematically computed for expectations in small-noise stochastic systems with additive noise?
• RQ2 What is the correct form of the Riccati equation that captures the fluctuation determinant for exit probabilities and mean first passage times?
• RQ3 How can sharp asymptotic estimates be extended to systems with invariant measures and infinite time horizons?
• RQ4 Can the method produce accurate, numerically stable estimates for rare event probabilities in infinite-dimensional stochastic PDEs?
• RQ5 How do the proposed Riccati-based formulas compare to direct Monte Carlo sampling in terms of accuracy and computational cost for small noise?

Key findings

• The paper derives explicit, numerically tractable formulas for the prefactor in the small-noise limit, enabling absolute-scale estimates of probabilities and expectations beyond exponential decay rates.
• The method successfully computes mean first passage times and exit probabilities in nonlinear, irreversible two-dimensional systems, with results validated against stochastic simulations.
• For the invariant measure, the approach yields accurate estimates of stationary probabilities and first-passage times in gradient and non-gradient diffusions, including in infinite-dimensional SPDEs.
• The Riccati equations are shown to be well-posed and solvable both forward and backward in time, with appropriate initial or final conditions for each type of estimate.
• Numerical results for a one-dimensional jump process and a linear SPDE with non-local forcing confirm the method’s accuracy and robustness, even in non-reversible and high-dimensional settings.
• The framework is efficient and scalable, providing reliable estimates in regimes where direct sampling is computationally infeasible due to rare event statistics.