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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
M. Abramowitz, I.A. Stegun (eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Reprint of the 1972 edition (Dover, New York, 1992)
H. Bercovici, V. Pata, Stable Laws and domains of attraction in free probability theory. With an appendix by Philippe Biane. Ann. Math. 149, 1023–1060 (1999)
H. Bercovici, D. Voiculescu, Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42, 733–773 (1993)
P. Biane, R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Prob. Theory Relat. Field 112, 373–409 (1998)
B. Bollobás, Linear Analysis, 2nd edn. (Cambridge University Press, Cambridge, 1999)
M. Capitaine, C. Donati-Martin, The Lévy area process for the free Brownian motion. J. Funct. Anal. 179, 153–169 (2001)
P. Di Francesco, O. Golinelli, E. Guitter, Meander, folding and arch statistics. Math. Comput. Modelling 26(8), 97–147 (1997)
F. Hiai, D. Petz, The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs, Vol. 77 (American Mathematical Society, Providence, RI, 2000)
M. Kac, On some connections between probability theory and differential and integral equations. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 189–215 (1950)
S.K. Lando, A. Zvonkin, Plane and projective meanders. Theor. Comput. Sci. 117, 227–241 (1993)
P. Lévy, Wiener’s random function, and other laplacian random functions. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 171–187 (1950)
T. Lyons, Differential equations driven by rough signals. Revista Matématica Iberoamericana 14, 215–310 (1998)
A. Nica, R. Speicher, Commutators of free random variables. Duke Math. J., 92(3), 553–592 (1998)
A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability (Cambridge University Press, Cambridge, 2006)
J. Ortmann, Large deviations for non-crossing partitions. arXiv preprint arXiv:1107.0208 (2011)
J. Ortmann, Random matrices, large deviations and reflected Brownian motion. PhD thesis, Warwick Mathematics Institute, 2011
R.P. Stanley, Enumerative Combinatorics, vol. 1, vol. 49 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1997)
L. Tolmatz, On the distribution of the square integral of the Brownian bridge. Ann. Probab. 30(1), 253–269 (2002)
N. Victoir, Lévy area for the free Brownian motion: existence and non-existence. J. Funct. Anal. 208, 107–121 (2004)
D.V. Voiculescu, Lectures on Free Probability Theory. No. 1738 in Lecture Notes in Mathematics (Lectures on Probability and Theory and Statistics) (Springer, 2000), pp. 283–349
D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables. CRM Monograph Series, vol. 1 (American Mathematical Society, Providence, RI, 1992)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-00321-4_17
Published:
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00320-7
Online ISBN: 978-3-319-00321-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)