Scaling limits for fractional polyharmonic Gaussian fields

01/31/2023

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by   Nicola De Nitti, et al.

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This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian (-Δ)^-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 (-Δ)^s under minimal regularity assumptions, which is also of independent interest.