# Sample Paths Estimates for Stochastic Fast-Slow Systems driven by Fractional Brownian Motion

## TL;DR

This paper investigates how additive fractional noise with H > 1/2 influences fast-slow systems, providing probabilistic bounds and neighborhood estimates, especially in systems with hyperbolic stable slow manifolds, with applications to climate modeling.

## Contribution

It extends sample path estimates to fractional Brownian motion driven systems, overcoming the lack of martingale methods and providing high-probability neighborhoods for such systems.

## Key findings

- High-probability neighborhoods containing the process are established.
- Exponential error estimates quantify the probability of system leaving the neighborhood.
- Application demonstrated in a climate modeling example with time-correlated noise.

## Abstract

We analyze the effect of additive fractional noise with Hurst parameter $H > \frac{1}{2}$ on fast-slow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood. We also illustrate our results in an example arising in climate modeling, where time-correlated noise processes have become of greater relevance recently.