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Some Identities in Law

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Local Times and Excursion Theory for Brownian Motion

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2088))

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Abstract

Several identities in law are shown to derive from excursion theory.

  • The law of the total local times process of the sum of reflecting Brownian motion and a multiple of its local time at 0 is shown to be that of a BESQ process, thus completing the Ray–Knight theorems. The Lévy–Khintchine formula for the BESQ laws is given.

  • The joint law of the maximum, minimum, and local time of a Brownian bridge is described, as well as the analogous law for Brownian motion.

  • Knight’s identity in law for the ratio of inverse local time divided by the square of the maximum up to that time is obtained.

  • Likewise, the Foldes–Revesz identity in law about the measure of levels spent by Brownian local times up to inverse local time is derived. Several identities in law involving Bessel processes up to last passage times are shown in the same manner.

  • The law of the Cauchy principal value of Brownian local times, considered up to inverse local time, or to an independent exponential time, is obtained. It has some parenthood with Lévy stochastic area formula.

  • The functional equation for the Riemann zeta function is shown to be closely related with a symmetry property of the law of the sum of two independent hitting times of BES(3).

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA

    Ju-Yi Yen

  2. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, Paris CX 05, France

    Marc Yor

Authors
  1. Ju-Yi Yen
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  2. Marc Yor
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Yen, JY., Yor, M. (2013). Some Identities in Law. In: Local Times and Excursion Theory for Brownian Motion. Lecture Notes in Mathematics, vol 2088. Springer, Cham. https://doi.org/10.1007/978-3-319-01270-4_11

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