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

TL;DR

This paper proves the convergence of the Hausdorff measure of level sets of smooth stationary Gaussian fields as the levels converge, using a novel divergence theorem approach instead of traditional Kac-Rice formulas.

Contribution

It introduces a new method to estimate the difference in level set measures via mean curvature integrals, providing a different perspective from existing techniques.

Findings

Hausdorff measure of level sets converges with level

Difference in measures can be estimated by $C^2$-fluctuations

Divergence theorem approach offers an alternative to Kac-Rice

Abstract

We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. 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.