Some Explicit Krein Representations of Certain Subordinators, Including the Gamma Process

Donati-Martin, Catherine; Yor, Marc

doi:10.2977/prims/1166642190

Cited by 16 publications

(28 citation statements)

References 27 publications

(38 reference statements)

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“…Furthermore, albeit not explicitly addressed in Bertoin has a similar interpretation where (R t , t > 0) is now replaced by a process (R (α,c) t , t > 0) whose inverse local time is distributed as a generalized gamma subordinator, that is, a subordinator whose Lévy density is specified by Cy −α−1 e −cy for y > 0. This interpretation may be deduced from Donati-Martin and Yor ( [10], see page 880 (1.c)), where R (α,c) equates with a downwards Bessel process with drift c.…”

Section: The Finite-dimensional Distribution Of Subordinators Of Bertmentioning

confidence: 88%

Dirichlet mean identities and laws of a class of subordinators

James¹

2010

Bernoulli

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An interesting line of research is the investigation of the laws of random variables known as Dirichlet means. However, there is not much information on interrelationships between different Dirichlet means. Here, we introduce two distributional operations, one of which consists of multiplying a mean functional by an independent beta random variable, the other being an operation involving an exponential change of measure. These operations identify relationships between different means and their densities. This allows one to use the often considerable analytic work on obtaining results for one Dirichlet mean to obtain results for an entire family of otherwise seemingly unrelated Dirichlet means. Additionally, it allows one to obtain explicit densities for the related class of random variables that have generalized gamma convolution distributions and the finite-dimensional distribution of their associated L\'{e}vy processes. The importance of this latter statement is that L\'{e}vy processes now commonly appear in a variety of applications in probability and statistics, but there are relatively few cases where the relevant densities have been described explicitly. We demonstrate how the technique allows one to obtain the finite-dimensional distribution of several interesting subordinators which have recently appeared in the literature.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ224 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Section: The Finite-dimensional Distribution Of Subordinators Of Bertmentioning

confidence: 88%

Dirichlet mean identities and laws of a class of subordinators

James¹

2010

Bernoulli

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“…Note that the tempered stable distribution (up to constants) arises in the theory of Vershik-Yor subordinator (see [23], and references therein). This section constructs a multifractal process based on the geometric tempered stable OU process.…”

Section: Log-tempered Stable Scenariomentioning

confidence: 99%

Multifractal scenarios for products of geometric Lévy-based stationary models

Denisov

Leonenko

2016

Stochastic Analysis and Applications

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“…In particular, the operator (m 2 − ∆) α/2 − m α is the Dirichlet-to-Neumann operator for the differential operator α −1 c α m α ∇ t,x (t(K α/2 (mt)) 2 ∇ t,x u(t, x)); here we set µ = m 2 . This calculation is due to [13] in probabilistic context.…”

Section: 5mentioning

confidence: 99%

Extension technique for complete Bernstein functions of the Laplace operator

Kwaśnicki

Mucha

2018

J. Evol. Equ.

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We discuss representation of certain functions of the Laplace operator ∆ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies (−∆) 1/2 , the square root of the d-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the (d + 1)-dimensional Laplace operator ∆ t,x in (0, ∞) × R d . Caffarelli and Silvestre extended this to fractional powers (−∆) α/2 , which correspond to operators ∇ t,x (t 1−α ∇ t,x ). We provide an analogous result for all complete Bernstein functions of −∆ using Krein's spectral theory of strings.Two sample applications are provided: a Courant-Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrödinger operators ψ(−∆) + V (x), as well as an upper bound for the eigenvalues of these operators. Here ψ is a complete Bernstein function and V is a confining potential.

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Some Explicit Krein Representations of Certain Subordinators, Including the Gamma Process

Abstract: We give a representation of the Gamma subordinator as a Krein functional of Brownian motion, using the known representations for stable subordinators and Esscher transforms.

Cited by 16 publications

References 27 publications

Dirichlet mean identities and laws of a class of subordinators

Dirichlet mean identities and laws of a class of subordinators

Multifractal scenarios for products of geometric Lévy-based stationary models

Extension technique for complete Bernstein functions of the Laplace operator

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