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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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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DOI: https://doi.org/10.1007/978-3-319-57805-7_13
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