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Large deviations of slow-fast systems driven by fractional Brownian motion

Siragan Gailus, Ioannis Gasteratos

TL;DR

The paper develops a large deviations framework for slow-fast stochastic systems driven by a fractional Brownian motion with $H>1/2$, under a homogenization regime where $\epsilon\to0$, $\eta\to0$, and $\sqrt{\eta}/\sqrt{\epsilon}\to0$. Employing a weak convergence/variational approach with controlled slow-fast dynamics and occupation measures, it proves an LDP with a rate function $S^H_{x_0}$ that, in certain regimes, takes an explicit non-variational form and exhibits a discontinuity as $H\to1/2^+$. The rate function is analyzed relative to the classical Freidlin–Wentzell theory, including a detailed limit as $H\to1/2^+$ and conditions under which the LDP rate remains continuous. The results advance the understanding of tail behavior in models with memory effects and offer a starting point for numerical rare-event methods in fractional-noise settings. The work also identifies avenues for extending the theory to general Gaussian inputs and alternative time-scale regimes.

Abstract

We consider a multiscale system of stochastic differential equations in which the slow component is perturbed by a small fractional Brownian motion with Hurst index $H>1/2$ and the fast component is driven by an independent Brownian motion. Working in the framework of Young integration, we use tools from fractional calculus and weak convergence arguments to establish a Large Deviation Principle in the homogenized limit, as the noise intensity and time-scale separation parameters vanish at an appropriate rate. Our approach is based in the study of the limiting behavior of an associated controlled system. We show that, in certain cases, the non-local rate function admits an explicit non-variational form. The latter allows us to draw comparisons to the case $H=1/2$ which corresponds to the classical Freidlin-Wentzell theory. Moreover, we study the asymptotics of the rate function as $H\rightarrow{1/2}^+$ and show that it is discontinuous at $H=1/2.$

Large deviations of slow-fast systems driven by fractional Brownian motion

TL;DR

The paper develops a large deviations framework for slow-fast stochastic systems driven by a fractional Brownian motion with , under a homogenization regime where , , and . Employing a weak convergence/variational approach with controlled slow-fast dynamics and occupation measures, it proves an LDP with a rate function that, in certain regimes, takes an explicit non-variational form and exhibits a discontinuity as . The rate function is analyzed relative to the classical Freidlin–Wentzell theory, including a detailed limit as and conditions under which the LDP rate remains continuous. The results advance the understanding of tail behavior in models with memory effects and offer a starting point for numerical rare-event methods in fractional-noise settings. The work also identifies avenues for extending the theory to general Gaussian inputs and alternative time-scale regimes.

Abstract

We consider a multiscale system of stochastic differential equations in which the slow component is perturbed by a small fractional Brownian motion with Hurst index and the fast component is driven by an independent Brownian motion. Working in the framework of Young integration, we use tools from fractional calculus and weak convergence arguments to establish a Large Deviation Principle in the homogenized limit, as the noise intensity and time-scale separation parameters vanish at an appropriate rate. Our approach is based in the study of the limiting behavior of an associated controlled system. We show that, in certain cases, the non-local rate function admits an explicit non-variational form. The latter allows us to draw comparisons to the case which corresponds to the classical Freidlin-Wentzell theory. Moreover, we study the asymptotics of the rate function as and show that it is discontinuous at
Paper Structure (21 sections, 28 theorems, 246 equations)

This paper contains 21 sections, 28 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Notation assumptions and preliminaries
  3. Function and measure spaces
  4. Fractional calculus and generalized Stieltjes integration.
  5. Fractional Brownian motion and Cameron-Martin spaces.
  6. Assumptions
  7. Weak convergence method and main results
  8. Tightness estimates for the controlled slow dynamics
  9. Limiting behavior of the controlled dynamics
  10. Proof of the Large Deviation Principle
  11. The Laplace Principle upper bound
  12. An equivalent form of the rate function and compactness of sublevel sets
  13. The Laplace Principle lower bound
  14. Case 1: $\mathbf{b=\sigma_2=0, \bar{\sigma}_1\neq 0}$
  15. Case 2: $\mathbf{\overline{QQ^T}(x)}$ is uniformly non-degenerate
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $T>0, u\in\mathcal{A}_N$ as in AN and $(X^{\epsilon,\eta,u}, Y^{\epsilon,\eta,u})$ solve the controlled system consys with initial conditions $(x_0,y_0)\in\mathcal{X}\times\mathcal{Y}$. Moreover, let $\Lambda_1:\mathcal{X}\times\mathcal{U}_1\times\mathcal{U}_2\times\mathcal{Y}\rightarrow\mathca where $\Psi$ is the unique strong solution of Poisson, $c_H$ as in cH and set $\Lambda=(\Lambda_1,\

Theorems & Definitions (73)

  • Definition 2.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Definition 3.1
  • Remark 7
  • Theorem 3.1
  • ...and 63 more