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Publication# The Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism

Abstract

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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Let X = {X(t); t ∈ RN} be a (N,d) fractional Brownian motion in Rd of index H ∈ (0,1). We study the local time of X for all temporal dimensions N and spatial dimensions d for which local time exist. We obtain two main results : R1. If we denote by Lx(I) the local time of X at x on I ⊂ RN, then there exists a positive finite constant c such that mφ(X-1(0) ∩ [0,1]N) = c L0([0,1]N), where φ(r) = rN-dH (log log 1/r)dH/N and mφ(E) is the Hausdorff φ-measure of E. This solves the problem of the relationship between the local time and the exact Hausdorff measure of zero set for X. R2. We refine results of Xiao (1997) for the local times of (N,d) fractional Brownian motion. We prove the law of iterated logarithm and global Hölder condition for the local time for our process. These results establish interesting properties which were only partially proved in the literature. This literature began with the work of Taylor and Wendel (1966) and Perkins (1981) for the first result; Kesten (1965) and Perkins (1981) for the second on the Brownian motion. It continued with several works of which that of Xiao (1997) on the locally nondeterministic processes with stationnary increments including the (N,d) fractional Brownian motion. An intermediate result is found to solve the case N > 1. We generalize the result of Kasahara et al. (1999) on the tail probability of local time.