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Regularity of the solutions to SPDEs in metric measure spaces

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Abstract

In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernel estimates. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is Hölder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4.

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Notes

  1. In [23] the \(\omega \) is missing in the exponent.

  2. We remark that there is a typo in [23], Lemma 5.2], namely in (ii) and (iii) the right hand side of the main condition on the parameters should read \(2-2\eta -\beta \) instead of \(2-2\eta -(\beta \vee \frac{d_S}{2})\).

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Authors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Elena Issoglio

  2. Institute of Mathematics, Friedrich Schiller University, 07743, Jena, Germany

    Martina Zähle

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  1. Elena Issoglio
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  2. Martina Zähle
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Correspondence to Martina Zähle.

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Issoglio, E., Zähle, M. Regularity of the solutions to SPDEs in metric measure spaces. Stoch PDE: Anal Comp 3, 272–289 (2015). https://doi.org/10.1007/s40072-015-0048-8

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