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

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.

arXiv comment27 pages, 5 figures

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• Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems
• Computing non-equilibrium trajectories by a deep learning approach
• Instantons, fluctuations and singularities in the supercritical stochastic nonlinear Schroedinger equation
• Noise equals control
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Synthetic Turbulence via an Instanton Gas Approximation
• Full distribution of the ground-state energy of potentials with weak disorder