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Journal of Statistical Physics

, Volume 168, Issue 2, pp 331–368 | Cite as

Model Spaces of Regularity Structures for Space-Fractional SPDEs

  • Nils Berglund
  • Christian Kuehn
Article
  • 426 Downloads

Abstract

We study model spaces, in the sense of Hairer, for stochastic partial differential equations involving the fractional Laplacian. We prove that the fractional Laplacian is a singular kernel suitable to apply the theory of regularity structures. Our main contribution is to study the dependence of the model space for a regularity structure on the three-parameter problem involving the spatial dimension, the polynomial order of the nonlinearity, and the exponent of the fractional Laplacian. The goal is to investigate the growth of the model space under parameter variation. In particular, we prove several results in the approaching subcriticality limit leading to universal growth exponents of the regularity structure. A key role is played by the viewpoint that model spaces can be identified with families of rooted trees. Our proofs are based upon a geometrical construction similar to Newton polygons for classical Taylor series and various combinatorial arguments. We also present several explicit examples listing all elements with negative homogeneity by implementing a new symbolic software package to work with regularity structures. We use this package to illustrate our analytical results and to obtain new conjectures regarding coarse-grained network measures for model spaces.

Keywords

Stochastic partial differential equations Regularity structures Fractional Laplacian Nonhomeomorphic rooted trees Subcriticality boundary 

Mathematics Subject Classification

60H15 35R11 (Primary) 05C05 82B20 (Secondary) 

Notes

Acknowledgements

We would like to thank Romain Abraham, Marie Albenque and Kilian Raschel for advice on the combinatorics of trees. CK has been supported by the VolkswagenStiftung via a Lichtenberg Professorship.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Laboratoire MAPMO, CNRS UMR 7349 and Fédération Denis Poisson FR 2964Université dOrléansOrléans Cedex 2France
  2. 2.Faculty of MathematicsTechnical University of Munich (TUM)Garching bei MünchenGermany

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