# Model Spaces of Regularity Structures for Space-Fractional SPDEs

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.

**Comments:**34 pages, 11 figures; preprint: comments and suggestions welcome

## Similar Publications

Parabolic integro-differential model Cauchy problem is considered in the scale of Lp -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some rough probability density function estimates of the associated Levy process are used as well. Read More

Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $d = \omega(\log^{16} n)$ and $d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d}$ converges weakly to the circular law in probability as $n \to \infty$. Read More

We consider the solution of a stochastic differential equation with additive
multidimensional fractional noise. In the case $\frac14

Graph bootstrap percolation is a variation of bootstrap percolation introduced by Bollob\'as. Let $H$ be a graph. Edges are added to an initial graph $G=(V,E)$ if they are in a copy of $H$ minus an edge, until no further edges can be added. Read More

We prove a stochastic averaging theorem for stochastic differential equations in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator ${\mathcal L}_x$. The theorem is proved under the assumption that ${\mathcal L}$ satisfies H\"ormander's bracket conditions, or more generally ${\mathcal L}$ is a family of Fredholm operators with sub-elliptic estimates. Read More

We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by $a \geq 0$, which creates a shock in the particle density of order $a t^{-1/3}$, $t$ the observation time. When starting from step initial data, we show that, in the double limit $\lim_{ a \to \infty}\lim_{t \to \infty}$ we recover the product limit law and the degeneration of the correlation length observed at shocks of order 1 in \cite{FN14}. This result is shown to apply to a general last-passage percolation model. Read More

The paper is devoted to the contribution in the Probability Theory of the well-known Soviet mathematician Alexander Yakovlevich Khintchine (1894-1959). Several of his results are described, in particular those fundamental results on the infinitely divisible distributions. Attention is paid also to his interaction with Paul Levy. Read More

Slicing a Voronoi tessellation in $\mathbb{R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in $\mathbb{R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. Read More

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. Read More

We prove the explicit formula for sofic and Rokhlin entropy of actions arising from some class of Gibbs measures. It provides a new set of examples with sofic entropy independent of sofic approximations. It is particularilly interresting, since in non-amenable case Rokhlin entropy was computed only in case of Bernoulli actions and for some examples with zero Rokhlin entropy. Read More