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

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 ex...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Schorlepp, Timo, Grafke, Tobias, Grauer, Rainer
Format: Article
Language:English
Subjects:
Calculus
Instantons
Mathematical analysis
Probability density functions
Stochastic systems
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2103.04887