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Amplitude equations for SPDEs driven by fractional additive noise with small hurst parameter

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2022

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Stochastics and Dynamics. World Scientific Publishing. 2022, 22(3), 2240013. eISSN 0219-4937. Available under: doi: 10.1142/S0219493722400135

Zusammenfassung

We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter H∈(0,1). Close to a change of stability measured with a small parameter ε, we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. Up to an error term bounded by a power of ε depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so-called amplitude equation, which describes the amplitude of the dominating pattern changing stability. In second order the approximation is given by a fast infinite-dimensional Ornstein–Uhlenbeck process. To this aim, we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters.

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510 Mathematik

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Amplitude equations, multiscale analysis, fractional Brownian motion, Young-type estimates

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ISO 690BLÖMKER, Dirk, Alexandra BLESSING-NEAMTU, 2022. Amplitude equations for SPDEs driven by fractional additive noise with small hurst parameter. In: Stochastics and Dynamics. World Scientific Publishing. 2022, 22(3), 2240013. eISSN 0219-4937. Available under: doi: 10.1142/S0219493722400135
BibTex
@article{Blomker2022Ampli-57941,
  year={2022},
  doi={10.1142/S0219493722400135},
  title={Amplitude equations for SPDEs driven by fractional additive noise with small hurst parameter},
  number={3},
  volume={22},
  journal={Stochastics and Dynamics},
  author={Blömker, Dirk and Blessing-Neamtu, Alexandra},
  note={Article Number: 2240013}
}
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