Moments of last exit times for Lévy processes

Sato, K

doi:10.1016/s0246-0203(03)00044-x

Cited by 14 publications

(24 citation statements)

References 9 publications

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“…We write p(1, x) = p(x). Then p(x) is of class C ∞ and satisfies the relation [33]. Let {X t } be a nondegenerate α-stable process on R d with 0 < α 2 and Lévy measure ν.…”

Section: Preliminariesmentioning

confidence: 99%

“…After Sato [29], [30], the set T has been studied in detail by Sato and Watanabe [33], [34]. In [33], they obtained a criterion for η ∈ T for a general transient Lévy process and discussed the relation with the moment of the last exit time from a half line for a one-dimensional random walk. This had been studied by Janson [16] f (x)(E x f (X s ))dx.…”

Section: Applications To Stable Processesmentioning

confidence: 99%

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Asymptotic estimates of multi-dimensional stable densities and their applications

Watanabe

2007

Trans. Amer. Math. Soc.

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Abstract. The relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed. In particular, the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of Pruitt and Taylor in 1969 are solved. The proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere, or discrete with the support being a finite set, or a mixture of such cases are obtained. Those results are applied to the moment of the last exit time from a ball and the Spitzer type limit theorem involving capacity for a multi-dimensional transient stable process.

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“…We write p(1, x) = p(x). Then p(x) is of class C ∞ and satisfies the relation [33]. Let {X t } be a nondegenerate α-stable process on R d with 0 < α 2 and Lévy measure ν.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Applications To Stable Processesmentioning

confidence: 99%

Asymptotic estimates of multi-dimensional stable densities and their applications

Watanabe

2007

Trans. Amer. Math. Soc.

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“…The proof is borrowed from [30,31]. Those papers deal only with processes on R d but the argument is general.…”

Section: Degree Of Transience and Moments Of Last Exit Timesmentioning

confidence: 99%

Degrees of Transience and Recurrence and Hierarchical Random Walks

2005

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The notion of degree and related notions concerning recurrence and transience for a class of Lévy processes on metric Abelian groups are studied. The case of random walks on a hierarchical group is examined with emphasis on the role of the ultrametric structure of the group and on analogies and differences with Euclidean random walks. Applications to separation of time scales and occupation times of multilevel branching systems are discussed. (2000): 60J15, 60J30, 60B15, 60F05, 60J80. Mathematics Subject Classifications

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“…where p t is the transition probability of Z [DGW2,SaW]. For simple symmetric random walk on the d-dimensional lattice Z d , it is wellknown that dimension 2 is the borderline for transience.…”

Section: A System Of Branching Random Walksmentioning

confidence: 99%

Hierarchical Equilibria of Branching Populations

Dawson

Gorostiza

Wakolbinger

2004

Electron. J. Probab.

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The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group ΩN consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit N → ∞ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls B (N) ℓ of hierarchical radius ℓ converge to a backward Markov chain on R+. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population. (2000): Primary 60J80; Secondary 60J60, 60G60. Mathematics Subject Classifications

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Moments of last exit times for Lévy processes

Cited by 14 publications

References 9 publications

Asymptotic estimates of multi-dimensional stable densities and their applications

Asymptotic estimates of multi-dimensional stable densities and their applications

Degrees of Transience and Recurrence and Hierarchical Random Walks

Hierarchical Equilibria of Branching Populations

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