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

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 all...

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Bibliographic Details
Published in:arXiv.org 2019-11
Main Authors: Eichinger, Katharina, Kuehn, Christian, Neamtu, Alexandra
Format: Article
Language:English
Subjects:
Brownian motion
Climate models
Estimates
Martingales
Random walk theory
Time correlation functions
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1905.06824