Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process

Freddy Bouchet, Roger Tribe, and Oleg Zaboronski
Phys. Rev. E 107, 034111 – Published 7 March 2023

Abstract

Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multidimensional case. This provides a large class of random dynamical systems with timescale separation for which an explicit sample-path large-deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multivariate Gaussian process and analyze its large-deviation functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large-deviation effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.

  • Figure
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  • Received 5 November 2021
  • Revised 8 June 2022
  • Accepted 30 January 2023

DOI:https://doi.org/10.1103/PhysRevE.107.034111

©2023 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
Statistical Physics & Thermodynamics

Authors & Affiliations

Freddy Bouchet*

  • Laboratoire de Physique ENS de Lyon and CNRS, 46 Alley d'Italie, F-69364 Lyon Cedex 07, France

Roger Tribe and Oleg Zaboronski

  • Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom

  • *freddy.bouchet @ ens-lyon.fr
  • r.p.tribe@warwick.ac.uk
  • o.v.zaboronski@warwick.ac.uk

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Issue

Vol. 107, Iss. 3 — March 2023

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