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.

翻译：本文研究分数阶高斯场，即协方差算子由分数阶拉普拉斯逆算子 $(-\Delta)^{-s}$ 给出的高斯场（特别地，我们考虑 $s >1$ 的情形）。我们定义了这些场的离散化格点形式，并证明了其标度极限——在最优贝索夫空间拓扑下（边界情形除外）——即为原始连续场。作为副产品，在维度 $d<2s$ 的情况下，我们证明了场的最大值的分布收敛性。证明的一个关键工具是在最小正则性假设下，对 $(-\Delta)^s$ 的自然有限差分格式给出的尖锐误差估计，该结果本身也具有独立意义。