# Model Spaces of Regularity Structures for Space-Fractional SPDEs

## TL;DR

This paper analyzes the structure and growth of model spaces for space-fractional SPDEs using regularity structures, providing new insights into their dependence on key parameters and introducing a software tool for explicit computations.

## Contribution

It introduces a detailed study of the parameter-dependent growth of model spaces for fractional SPDEs within the regularity structures framework, including explicit examples and software implementation.

## Key findings

- Model spaces grow universally near subcriticality limits.
- Fractional Laplacian acts as a suitable singular kernel.
- New software package enables explicit enumeration of model elements.

## 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.