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Register a new userThe Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism
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Cheuk Yin Lee
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We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space-time white noise or colored noise.
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For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area $R^2$. The mean number of components is known to be of order $R^2$ for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels $ell$, these random variables have fluctuations of order at least $R$, and hence variance of order at least $R^2$. In particular, this holds for excursion sets when $ell$ is in some neighbourhood of zero, and it holds for excursion/level sets when $ell$ is sufficiently large. We prove stronger fluctuation lower bounds of order $R^alpha$, $alpha in [1,2]$, in the case that the spectral density has a singularity at the origin. Finally, we show that the number of excursion/level sets for the Random Plane Wave at certain levels has fluctuations of order at least $R^{3/2}$, and hence variance of order at least~$R^3$. We expect that these bounds are of the correct order, at least for generic levels.
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