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(S)PDE on Fractals and Gaussian Noise

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Recent Developments in Fractals and Related Fields (FARF3 2015)

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Abstract

In the first part of this paper we give a survey on results from Hinz and Zähle (Potential Anal 36:483–515, 2012) and Issoglio and Zähle (Stoch PDE Anal Comput 3:372–389, 2015) for nonlinear parabolic (S)PDE on certain metric measure spaces of spectral dimensions less than 4 with applications to fractals. We consider existence, uniqueness, and fractional regularity properties of mild function solutions in the pathwise sense. In the second part we apply this to the special case of fractal Laplace operators as generators and Gaussian random noises.

Furthermore, we show that random space-time fields Y (t, x) like fractional Brownian sheets with Hurst exponents H in time and K in space on general Ahlfors regular compact metric measure spaces X possess a modification whose sample paths are elements of C α([0, t 0], C β(X)) for all α < H and β < K. This is used in the above special case of SPDE on fractals.

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Notes

  1. 1.

    In [16] the w is missing in the exponent.

  2. 2.

    We remark that there is a typo in [16, Lemma 5.2], namely in (ii) and (iii) the right-hand side of the main condition on the parameters should read 2 − 2η −β instead of \(2 - 2\eta - (\beta \vee \frac{d_{S}} {2} )\).

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Authors and Affiliations

  1. Institute of Mathematics, Friedrich Schiller University, 07737, Jena, Germany

    Martina Zähle

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  1. Martina Zähle
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Correspondence to Martina Zähle .

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Editors and Affiliations

  1. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Julien Barral

  2. Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-Est, UPEMLV, UPEC, CNRS, Créteil, France

    Stéphane Seuret

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Zähle, M. (2017). (S)PDE on Fractals and Gaussian Noise. In: Barral, J., Seuret, S. (eds) Recent Developments in Fractals and Related Fields. FARF3 2015. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-57805-7_13

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