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

Freddy Bouchet1, Roger Tribe2, Oleg Zaboronski2

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This study introduces a method to calculate large deviation rates for random processes driven by quadratic forms of Gaussian processes. This enables finding explicit functionals for metastability in systems with timescale separation.

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Area of Science:

  • Stochastic processes
  • Statistical physics
  • Dynamical systems

Background:

  • Physical models often involve random processes with complex driving forces.
  • Understanding large deviation principles is crucial for analyzing system stability and rare events.
  • Timescale separation is a common feature in many physical systems, from fluid dynamics to climate science.

Purpose of the Study:

  • To develop a general method for computing sample-path large deviation rate functions for a specific class of random processes.
  • To analyze the emergence of metastability in systems with timescale separation due to noise.
  • To construct instanton trajectories connecting metastable states in a model system.

Main Methods:

  • Utilizing the large domain size asymptotic of Fredholm determinants.
  • Applying Widom's theorem to generalize the Szegő-Kac formula to multidimensional cases.
  • Constructing and analyzing a specific example of a slow degree of freedom driven by a fast Gaussian process.

Main Results:

  • An analytical method for computing rate functions describing sample-path large deviations was derived.
  • A class of random dynamical systems with timescale separation was identified for which explicit large-deviation functionals can be found.
  • A model system demonstrated that noise can induce metastability, leading to multiple effective potential fixed points.

Conclusions:

  • The developed method provides explicit sample-path large-deviation functionals for a broad range of random dynamical systems with timescale separation.
  • Noise is shown to be a key factor in creating metastable states, even in systems with a single stable state in the absence of noise.
  • The derived rate functions allow for the construction of instanton trajectories, offering insights into the dynamics between metastable states.