Level Sets of Additive Lévy Processes

Khoshnevisan, Davar; Xiao, Yimin

doi:10.1214/aop/1020107761

Cited by 39 publications

(66 citation statements)

References 49 publications

(50 reference statements)

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“…It is quite standard to show that Among other things, we will show in this paper that, when (1.3) holds, the proper setting for the analysis of question (1.1) is potential theory and its various connections to the random field X, as well as energy that we will describe below. Various aspects of the potential theory of multiparameter processes have been treated in Evans (1987a, b), Fitzsimmons and Salisbury (1989), Hirsch (1995), Hirsch and Song (1995a, b) and Khoshnevisan and Xiao (2002a).…”

Section: X(t) = X 1 (T 1 ) + · · · + X N (T N )mentioning

confidence: 99%

See 1 more Smart Citation

Measuring the range of an additive Lévy process

Khoshnevisan¹,

Xiao²,

Zhong

2003

Ann. Probab.

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111

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The primary goal of this paper is to study the range of the random field X(t) = N j =1 X j (t j ), where X 1 , . . . , X N are independent Lévy processes in R d .To cite a typical result of this paper, let us suppose that i denotes the Lévy exponent of X i for each i = 1, . . . , N. Then, under certain mild conditions, we show that a necessary and sufficient condition for X(R N + ) to have positive d-dimensional Lebesgue measure is the integrability of the function. This extends a celebrated result of Kesten and of Bretagnolle in the one-parameter setting. Furthermore, we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion. This extends a theorem of Hawkes to the present random fields setting and completes the analysis of local times for additive Lévy processes initiated in a companion by paper Khoshnevisan, Xiao and Zhong.

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Section: X(t) = X 1 (T 1 ) + · · · + X N (T N )mentioning

confidence: 99%

“…The following key fact is borrowed from Khoshnevisan and Xiao (2002a), which we reproduce for the sake of completeness. (A) s j for all j = 1, .…”

Section: Proof Temporarily Let µ T Denote the Distribution Of − X(t)mentioning

confidence: 99%

Measuring the range of an additive Lévy process

Khoshnevisan¹,

Xiao²,

Zhong

2003

Ann. Probab.

Self Cite

111

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“…In Theorem 1.1 of Khoshnevisan and Xiao (2002), it is proved that (ii) in Corollary 3.7 holds with positive probability. It can be strengthened to a probability 1 result by using the conditional Borel-Cantelli lemma.…”

Section: And Only If F Carries a Finite Measure Of Finite Energymentioning

confidence: 99%

Additive Lévy Processes: Capacity and Hausdorff Dimension

Khoshnevisan

Xiao

2004

Fractal Geometry and Stochastics III

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Abstract. This is a survey on recently-developed potential theory of additive Lévy processes and its applications to fractal geometry of Lévy processes.Additive Lévy processes arise naturally in the studies of the Brownian sheet, intersections of Lévy processes and so on. We first summarize some recent results on the novel connections between an additive Lévy process X in R d , and a natural class of energy forms and their corresponding capacities. We then apply these results to study the Hausdorff dimension of the range and self-intersections of an ordinary Lévy process, solving several long-standing problems in the folklore of the theory of Lévy processes. We also list several open problems in this area. Mathematics Subject Classification (2000). Primary 60J25, 28A80; Secondary 60G51, 60G17.

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“…Dalang and Nualart (2004) have recently extended the methods of Khoshnevisan and Shi (1999) and proved similar results for the solution of a system of d nonlinear hyperbolic stochastic partial differential equations with two variables. In this context, we also mention that Khoshnevisan and Xiao (2002, 2003 and Khoshnevisan, Xiao and Zhong (2003) have established systematic potential theoretical results for additive Lévy processes in R d . The arguments in the aforementioned work rely on the multiparameter martingale theory; we refer to Khoshnevisan (2002) for more information on the latter as well as on potential theory of random fields.…”

Section: Remark 72 In the Critical Case Whenmentioning

confidence: 82%

Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao¹

Lecture Notes in Mathematics

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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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Level Sets of Additive Lévy Processes

Cited by 39 publications

References 49 publications

Measuring the range of an additive Lévy process

Measuring the range of an additive Lévy process

Additive Lévy Processes: Capacity and Hausdorff Dimension

Sample Path Properties of Anisotropic Gaussian Random Fields

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