Calculus of Variations and Geometric Measure Theory

N. De Nitti - F. Schweiger

Scaling limits for fractional polyharmonic Gaussian fields

created by denitti on 31 Jan 2023
modified on 01 Feb 2023


Submitted Paper

Inserted: 31 jan 2023
Last Updated: 1 feb 2023

Year: 2023

ArXiv: 2301.13781 PDF


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 -- are the original continuous fields. As a byproduct, in dimension $d<2s$, we prove the convergence in distribution of the maximum of the fields as well. 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.