Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems

TL;DR

This paper develops a method to accurately compute the fluctuations around instantons in stochastic systems, improving the estimation of tail probabilities of rare events by including prefactor contributions.

Contribution

It introduces a Gel'fand-Yaglom based recursive approach to evaluate fluctuation determinants around instantons in stochastic dynamics.

Findings

Method accurately predicts fluctuation prefactors in turbulence models.

Predictions agree with direct sampling results.

Enhances instanton calculus for rare event probability estimation.

Abstract

In recent years, instanton calculus has successfully been employed to estimate tail probabilities of rare events in various stochastic dynamical systems. Without further corrections, however, these estimates can only capture the exponential scaling. In this paper, we derive a general, closed form expression for the leading prefactor contribution of the fluctuations around the instanton trajectory for the computation of probability density functions of general observables. The key technique is applying the Gel'fand-Yaglom recursive evaluation method to the suitably discretized Gaussian path integral of the fluctuations, in order to obtain matrix evolution equations that yield the fluctuation determinant. We demonstrate agreement between these predictions and direct sampling for examples motivated from turbulence theory.