Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields

Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

“…To prove the above integral is finite, we observe that for any 0 < γ < min{d, N j =1 1 33) where N j =N+1 1 H j := 0. In the following, we will only consider the case of k = 1, the remaining cases are simpler because they require less steps of integration using Lemma 9.…”

“…Related to these problems, we mention that the Hausdorff measure functions for the range and graph of an (N, d)-fractional Brownian motion X have been obtained by Talagrand [29] and Xiao [33,34]; and the exact packing measure functions for X([0, 1] N ) have been studied by Xiao [32,35]. Their methods are useful for studying the fractional Brownian sheet B H as well.…”

Sample Path Properties of Anisotropic Gaussian Random Fields

“…Pitt (1978)]. As shown by Xiao (1997bXiao ( , 2007 and Shieh and Xiao (2006), many sample path properties of such Gaussian random fields are similar to those of fractional Brownian motion.…”

“…Sample path properties of such Gaussian random fields have been studied widely. See Xiao (1997bXiao ( , 2007, Shieh and Xiao (2006) Many Gaussian random fields can be constructed by choosing the spectral measures appropriately. For example, if we consider the spectral density …”

Fractional Brownian Motion

“…Consider x < x 0 , λ = x 0 /x > 1. The self-similarity of ξ(x) yields P (G(0, 1) ∈ dx) = P (G(0, λ) ∈ λdx) ≤ P (G(0, 1) ∈ λdx) = ψ(x 0 ) x 0 x dx (22) Here, G(a, b) is the leftmost position of the supremum of {ξ(x), x ∈ (a, b)}. Consequently, the distribution of G(0, 1) is absolutely continuous in (0, x 0 ).…”