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.
Highlights
Quantifying the probability of rare events is extraordinarily difficult: they are usually too rare to be efficiently observed or sampled, and at the same time too important to be ignored
Even though the instanton method is well established in the literature in order to estimate observable probability density function (PDF) of stochastic ordinary differential equations (SDEs) in a suitable large deviation limit, general procedures to obtain sharper estimates for these PDFs by including the full prefactor Z at leading order have not been investigated systematically in this context up until now
Apart from the unwieldy discretized expressions that we encountered in the derivation of our main result, our approach consists of a straightforward and conceptually simple evaluation of the Gaussian path integral that is obtained by expanding the action to second order around the instanton trajectory
Summary
Quantifying the probability of rare events is extraordinarily difficult: they are usually too rare to be efficiently observed or sampled, and at the same time too important to be ignored. We focus on the second step and develop a general formalism to compute the contributions of quadratic fluctuations around the instanton solution to the path integral for the evaluation of PDFs. We will present our approach for general finite dimensional Langevin equations, but with the focus that the developed methods are (in particular numerically) applicable to large systems of stochastic ordinary differential equations (SDEs) and to stochastic partial differential equations (SPDEs) relevant in fluid and plasma turbulence (e.g. Burgers, Navier–Stokes and the magnetohydrodynamic equations). The main technical issues that we address are the calculation of the marginal distribution by performing an appropriate integral over all permitted boundary conditions of the fluctuations, and the impact of the discretization of the path integral on the fluctuation matrix and its determinant in particular This leads to equations of the Gel’fand–Yaglom type, which can be linearized by a Radon transformation.
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