On convergence of volume of level sets of stationary smooth Gaussian fields

2023/02/20 by Dmitry Beliaev, Beliaev, Dmitry, Akshay Hegde +1 · 1 citation
Mathematics · #53A07 (Secondary) #60G60 (Primary) 60G15 #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2302.10265

openalex publication_date 2023/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields f₁, f₂ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of C²-fluctuations of the field F=f₁-f₂. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.

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