Model Spaces of Regularity Structures for SpaceFractional SPDEs
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 threeparameter 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 coarsegrained network measures for model spaces.
 Publication:

Journal of Statistical Physics
 Pub Date:
 July 2017
 DOI:
 10.1007/s1095501718013
 arXiv:
 arXiv:1701.03066
 Bibcode:
 2017JSP...168..331B
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Combinatorics;
 60H15;
 35R11;
 05C05;
 82B20
 EPrint:
 36 pages, 11 figures