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Journal ArticleDOI

Density factorizations for brownian motion, meander and the three-dimensional bessel process, and applications

01 Sep 1984-Journal of Applied Probability (Cambridge University Press (CUP))-Vol. 21, Iss: 3, pp 500-510
About: This article is published in Journal of Applied Probability.The article was published on 1984-09-01. It has received 146 citations till now. The article focuses on the topics: Meander (mathematics) & Bessel process.
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BookDOI
01 Jan 2006
TL;DR: In this paper, the Brownian forest and the additive coalescent were constructed for random walks and random forests, respectively, and the Bessel process was used for random mappings.
Abstract: Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

1,371 citations


Additional excerpts

  • ...These descriptions are read from [435, 208]....

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Journal ArticleDOI
TL;DR: In this article, a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles is presented.
Abstract: As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland–Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish–Chandra (Itzykson–Zuber) formula of integral over unitary group is established.

168 citations


Cites background from "Density factorizations for brownian..."

  • ...This is regarded as a multivariate version of the Imhof relation in the probability theory [25], since it implies the absolute continuity in distribution of the temporally homogeneous process Y(t) and the inhomogeneous process X(t) in [0, T ], but from the viewpoint of random matrix theory the important consequence of this equality is the fact that the process X(t) exhibits a transition in distribution from the eigenvalue statistics of GUE to that of GOE and thus the GOE distribution is realized at the final time t = T ....

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  • ...When N = 1 and (ν, κ) = (1/2, 1), this proposition gives the Imhof relation between the Brownian meander and the three-dimensional Bessel process [25]....

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Journal ArticleDOI
TL;DR: Arguin et al. as mentioned in this paper showed that the extremal point process of branching Brownian motion is a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process.
Abstract: It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, A branching random walk seen from the tip, 2010, Poissonian statistics in the extremal process of branching Brownian motion, 2010; Arguin et al., The extremal process of branching Brownian motion, 2011). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (The extremal process of branching Brownian motion, 2011).

143 citations

Journal ArticleDOI
TL;DR: In this paper, a self-similar Markov process X is constructed via its associated entrance law, which can be viewed as X conditioned never to hit 0, and then the process is constructed similarly to the way in which the Brownian excursion measure is constructed through the law of a Bessel(3) process.
Abstract: Let ξ be a real-valued Levy process that satisfies Cramer's condition, and X a self-similar Markov process associated with ξ via Lamperti's transformation. In this case, X has 0 as a trap and satisfies the assumptions set out by Vuolle-Apiala. We deduce from the latter that there exists a unique excursion measure \exc, compatible with the semigroup of X and such that \exc(X0+>0)=0. Here, we give a precise description of \exc via its associated entrance law. To this end, we construct a self-similar process X atural, which can be viewed as X conditioned never to hit 0, and then we construct \exc similarly to the way in which the Brownian excursion measure is constructed via the law of a Bessel(3) process. An alternative description of \exc is given by specifying the law of the excursion process conditioned to have a given length. We establish some duality relations from which we determine the image under time reversal of \exc.

129 citations

References
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