Local Nondeterminism and Hausdorff Dimension

Monrad, Ditlev; Pitt, Loren D.

doi:10.1007/978-1-4684-6751-2_12

Cited by 36 publications

(52 citation statements)

References 19 publications

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“…We also establish an explicit formula for the Hausdorff dimension of the image X(E) in terms of the generalized Hausdorff dimension of E (with respect to an appropriate metric) and the Hurst index H. Moreover, when H = α [see below for the notation], we prove the following uniform Hausdorff dimension result for the images of X: If N ≤ αd, then with probability one, dim H X(E) = 1 α dim H E for all Borel sets E ⊆ (0, ∞) N . (1.4) This extends the previous results of Monrad and Pitt (1987), Mountford (1989) and Khoshnevisan, Wu and Xiao (2006) for fractional Brownian motion and the Brownian sheet, respectively, and is another application of the strong local nondeterminism. In Section 7, we determine the Hausdorff and packing dimensions of the level sets, and establish estimates on the hitting probabilities of Gaussian random fields X satisfying Conditions (C1) and (C2).…”

Section: Introductionsupporting

confidence: 85%

“…The proof of Theorem 6.13 is reminiscent to those in Monrad and Pitt (1987), Khoshnevisan, Wu and Xiao (2006) and Wu and Xiao (2007). The key step is to apply Condition (C3) or (C3 ) to prove the following lemma.…”

Section: Theorem 613 Let X = {X(t) T ∈ R N } Be As In Theorem 611 mentioning

confidence: 91%

“…Here are some partial answers. If in addition to Conditions (C1) and (C2), we also assume Condition (C3) or (C3 ) holds and H 1 = H 2 = · · · = H N , then one can modify the proofs in Monrad and Pitt (1987) to show that the answer to the above question is affirmative. In general, it can be proved that, when N =1 1 H > d, the upper bound in (7.32) holds almost surely for all Borel sets F ⊆ R d .…”

Section: Theorem 76 Assume That An (N D)-gaussian Random Field X = mentioning

confidence: 99%

“…As such, they are useful in studying the various fractal properties of level sets and inverse images of the vector field X. In this regard, we refer to Berman (1972), Ehm (1981), Rosen (1984), Monrad and Pitt (1987) and Xiao (1997b). First we consider the existence of the local times of Gaussian random fields.…”

Section: Local Times and Their Joint Continuitymentioning

confidence: 99%

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Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao¹

Lecture Notes in Mathematics

141

227

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Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation.This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d and with parameters H 1 , . . . , H N . Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly.Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random fields. An important tool for our study is the various forms of strong local nondeterminism.

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Section: Introductionsupporting

confidence: 85%

Section: Theorem 613 Let X = {X(t) T ∈ R N } Be As In Theorem 611 mentioning

confidence: 91%

Section: Theorem 76 Assume That An (N D)-gaussian Random Field X = mentioning

confidence: 99%

Section: Local Times and Their Joint Continuitymentioning

confidence: 99%

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Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao¹

Lecture Notes in Mathematics

141

227

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“…Many authors have studied the sample path properties of fractional Brownian motion. See Adler [1], Kahane [5], Monrad and Pitt [10], Pitt [11], Rosen [12], Talagrand [13], Xiao [17] [18], just to mention a few.…”

Section: Introductionmentioning

confidence: 99%