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Séminaire de Probabilités XLV pp 433–458Cite as

Functionals of the Brownian Bridge

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Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2078))

Abstract

We discuss the distributions of three functionals of the free Brownian bridge: its L2-norm, the second component of its signature and its Lévy area. All of these are freely infinitely divisible. Two representations of the free Brownian bridge as series of free semicircular random variables are introduced and used. These are analogous to the Fourier representations of the classical Brownian bridge due to Lévy and Kac and the latter extends to all semicircular processes.

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Acknowledgements

The author would like to thank his PhD advisor, Neil O’Connell for his advice and support in the preparation of this paper. We also thank Philippe Biane for helpful discussions and suggestions and the anonymous referee whose detailed comments have lead to a much improved version of the paper.

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Authors and Affiliations

  1. Warwick Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

    Janosch Ortmann

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  1. Janosch Ortmann
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Correspondence to Janosch Ortmann .

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Editors and Affiliations

  1. , Laboratoire de Mathématiques, Université de Versailles-St Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Ortmann, J. (2013). Functionals of the Brownian Bridge. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_17

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