• Corpus ID: 232146916

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

@inproceedings{Grafke2021SharpAE,
title={Sharp Asymptotic Estimates for Expectations, Probabilities, and Mean First Passage Times in Stochastic Systems with Small Noise},
author={Tobias Grafke and Tobias Schafer and Eric Vanden-Eijnden},
year={2021}
}
• Published 8 March 2021
• Physics, Mathematics
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…
1 Citations

## Figures from this paper

Path integral derivation and numerical computation of large deviation prefactors for non-equilibrium dynamics through matrix Riccati equations
• Physics
• 2021
For many non-equilibrium dynamics driven by small noise, in physics, chemistry, biology, or economy, rare events do matter. Large deviation theory then explains that the leading order term of the

## References

SHOWING 1-10 OF 50 REFERENCES
Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools
• Computer Science, Mathematics
• 2017
The theoretical and computational aspects behind differential equations with multiplicative noise, Markov jump processes, and systems with fast and slow degrees of freedom are reviewed, and an algorithm that simplifies the geometric minimum action method to minimize the action in the space of arc-length parametrized curves is proposed.
Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times
• Mathematics
• 2004
We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form 1 +rF(·)r on R d or subsets of R d , whereF is a smooth
Rare Event Simulation of Small Noise Diffusions
• Mathematics
• 2012
An important sampling method for certain rare event problems involving small noise diffusions is proposed. Standard Monte Carlo schemes for these problems behave exponentially poorly in the small
Large fluctution for a non linear heat equation with noise
• Mathematics, Physics
• 1982
Studies a nonlinear heat equation in a finite interval of space subject to a white noise forcing term. The equation without the forcing term exhibits several equilibrium configurations, two of which
Minimum Action Method for the Study of Rare Events
• Physics
• 2004
The least-action principle from the Wentzell-Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a
The instanton method and its numerical implementation in fluid mechanics
• Mathematics, Physics
• 2015
A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent
An Eyring-Kramers law for the stochastic Allen-Cahn equation in dimension two
• Mathematics, Physics
• 2016
We study spectral Galerkin approximations of an Allen--Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon}$. We introduce a Wick
Exact Stochastic Simulation of Coupled Chemical Reactions
There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly
Limiting Exit Location Distributions in the Stochastic Exit Problem
• Mathematics, Physics
SIAM J. Appl. Math.
• 1997
It is shown using formal methods that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions.
Generalisation of the Eyring–Kramers Transition Rate Formula to Irreversible Diffusion Processes
• Mathematics, Physics
• 2015
In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius’ law. The Eyring–Kramers formula