Noise correction of large deviations with anomalous scaling

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.

v1: 13 pages, 5 figures. v2: minor typos corrected, close to published version

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• The large deviation approach to statistical mechanics
• Large Deviations in Single File Diffusion
• Current fluctuations in stochastic systems with long-range memory
• Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
• Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems

Cited by in corpus (4)

• Anomalous dynamical scaling determines universal critical singularities
• Universal singularities of anomalous diffusion in the Richardson class
• Symmetries and zero modes in sample path large deviations
• Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions