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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.13781 |
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| _version_ | 1866913894034833408 |
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| author | De Nitti, Nicola Schweiger, Florian |
| author_facet | De Nitti, Nicola Schweiger, Florian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13781 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scaling limits for fractional polyharmonic Gaussian fields De Nitti, Nicola Schweiger, Florian Probability Numerical Analysis Analysis of PDEs 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. |
| title | Scaling limits for fractional polyharmonic Gaussian fields |
| topic | Probability Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2301.13781 |