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Setvalued Dynamical Systems for Stochastic Evolution Equations Driven by Fractional Noise

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Abstract

We consider Hilbert-valued evolution equations driven by Hölder paths with Hölder index greater than 1/2, which includes the case of fractional noises with Hurst parameters in (1/2,1). The assumptions of the drift term will not be enough to ensure the uniqueness of solutions. Nevertheless, adopting a multivalued setting, we will prove that the set of all solutions corresponding to the same initial condition generates a (multivalued) nonautonomous dynamical system \(\Phi \). Finally, to prove that \(\Phi \) is measurable (and hence a (multivalued) random dynamical system), we need to construct a new metric dynamical system that models the noise with the property that the set space is separable.

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Acknowledgements

M.J. Garrido-Atienza was partially supported by FEDER and Spanish Ministerio de Economía y Competitividad, project MTM2015-63723-P and by Junta de Andalucía under Proyecto de Excelencia. J. Valero was partially supported by FEDER and Spanish Ministerio de Economía y Competitividad, projects MTM2015-63723-P and MTM2016-74921-P.

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Authors and Affiliations

  1. Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Avda. Reina Mercedes, s/n, 41012, Seville, Spain

    M. J. Garrido-Atienza

  2. Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 77043, Jena, Germany

    B. Schmalfuss

  3. Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. de la Universidad, s/n, 03202, Elche, Spain

    J. Valero

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  1. M. J. Garrido-Atienza
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  2. B. Schmalfuss
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  3. J. Valero
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Correspondence to B. Schmalfuss.

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Garrido-Atienza, M.J., Schmalfuss, B. & Valero, J. Setvalued Dynamical Systems for Stochastic Evolution Equations Driven by Fractional Noise. J Dyn Diff Equat 34, 79–105 (2022). https://doi.org/10.1007/s10884-019-09811-9

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  • DOI: https://doi.org/10.1007/s10884-019-09811-9

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