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

TL;DR

This paper establishes a large deviation principle for slow-fast stochastic systems driven by fractional Brownian motion with Hurst index greater than 1/2, using Young integration and fractional calculus, and compares it to classical theory.

Contribution

It introduces a novel large deviation framework for multiscale systems driven by fractional Brownian motion, including explicit rate functions and their asymptotic behavior as H approaches 1/2.

Findings

Large deviation principle established for H>1/2.

Explicit non-variational rate function derived in certain cases.

Discontinuity of the rate function at H=1/2 observed.

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 $H=1/2.$