Noise correction of large deviations with anomalous scaling

TL;DR

This paper calculates the probability distribution of time-integrated moments of the Ornstein-Uhlenbeck process, including Gaussian corrections, revealing how anomalous large deviations arise and why standard methods often fail.

Contribution

It provides a path integral approach with Gaussian prefactors for anomalous large deviations, extending previous instanton results and offering new insights into their creation.

Findings

Gaussian prefactor refines low-noise approximation

Defines an instanton variance for anomalous deviations

Explains failure of standard large deviation methods

Abstract

We present a path integral calculation of the probability distribution associated with the time-integrated moments of the Ornstein-Uhlenbeck process that includes the Gaussian prefactor in addition to the dominant path or instanton term obtained in the low-noise limit. The instanton term was obtained recently [D. Nickelsen, H. Touchette, Phys. Rev. Lett. 121, 090602 (2018)] and shows that the large deviations of the time-integrated moments are anomalous in the sense that the logarithm of their distribution scales nonlinearly with the integration time. The Gaussian prefactor gives a correction to the low-noise approximation and leads us to define an instanton variance giving some insights as to how anomalous large deviations are created in time. The results are compared with simulations based on importance sampling, extending our previous results based on direct Monte Carlo simulations. We conclude by explaining why many of the standard analytical and numerical methods of large deviation theory fail in the case of anomalous large deviations.