Scaling limits for fractional polyharmonic Gaussian fields

TL;DR

This paper investigates the scaling limits of fractional Gaussian fields, demonstrating their convergence to continuous fields under lattice discretization and establishing maximum distribution convergence in certain dimensions.

Contribution

It introduces a lattice discretization for fractional Gaussian fields and proves their convergence to the continuous fields in the optimal Besov space topology.

Findings

Scaling limits match original continuous fields

Convergence of the maximum in dimension d<2s

Sharp error estimates for finite difference schemes

Abstract

This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian $(-\Delta)^{-s}$ (where, in particular, we include the case $s >1$). We define a lattice discretization of these fields and show that their scaling limits -- with respect to the optimal Besov space topology (up to an endpoint case) -- are the original continuous fields. As a byproduct, in dimension $d<2s$, we prove the convergence in distribution of the maximum of the fields. A key tool in the proof is a sharp error estimate for the natural finite difference scheme for $(-\Delta)^s$ under minimal regularity assumptions, which is also of independent interest.