The Annals of Probability

Random Brownian scaling identities and splicing of Bessel processes

Jim Pitman and Marc Yor

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Abstract

An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of Brownian excursions, due to Williams and Bismut, each involving back-to-back splicing of fragments of two independent three-dimensional Bessel processes. Generalizations of both splicing constructions are described, which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.

Article information

Source
Ann. Probab. Volume 26, Number 4 (1998), 1683-1702.

Dates
First available: 31 May 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1022855878

Mathematical Reviews number (MathSciNet)
MR1675059

Digital Object Identifier
doi:10.1214/aop/1022855878

Zentralblatt MATH identifier
0937.60079

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G18: Self-similar processes 60J60: Diffusion processes [See also 58J65]

Keywords
Brownian bridge Brownian excursion Brownian scaling path transformation Williams' decomposition local time Bessel process range process

Citation

Pitman, Jim; Yor, Marc. Random Brownian scaling identities and splicing of Bessel processes. The Annals of Probability 26 (1998), no. 4, 1683--1702. doi:10.1214/aop/1022855878. http://projecteuclid.org/euclid.aop/1022855878.


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