Brownian motion, reflection groups and Tanaka formula. (English) Zbl 1263.60074

Demni, Nizar; Lépingle, Dominique

Rend. Semin. Mat. Univ. Padova 127, 41-55 (2012).

It is proved that a projection of a finite dimensional Wiener process (Brownian motion) on a closed Weyl chamber – the term from finite reflection group theory – is also a Wiener process reflected from the walls of the chamber. A decomposition of this reflected process is given, which may be regarded as a multidimensional extension of Tanaka’s formula. The links to local times at the boundary are established. A brief introduction to finite reflection group theory is provided.

Reviewer: Alexander Yu. Veretennikov (Leeds)

MSC:

60J65	Brownian motion
60G44	Martingales with continuous parameter
60J55	Local time and additive functionals

Keywords:

Brownian motion; reflection; Tanaka’s formula; local times

References:

[1]	P. BIANE - P. BOUGEROL - N. O’CONNELL, Littelman paths and Brownian paths. Duke Math. J., 130 (2004), pp. 127-167.
[2]	N. BOURBAKI, EÂleÂments de matheÂmatiques: Groupes et algeÁbres de Lie. Chapitres 4-6, Hermann, Paris 1968.
[3]	O. CHYBIRYAKOV, Processus de Dunkl et relation de Lamperti. Ph. D. Thesis, Universite de Paris VI, 2006. · Zbl 1094.60035
[4]	O. CHYBIRYAKOV - L. GALLARDO - M. YOR, Dunkl processes and their radial parts relative to a root system. Travaux en cours, 71 (Hermann 2008), pp. 113-197.
[5]	N. DEMNI, A guided tour in the world of radial Dunkl processes. Travaux en cours, 71 (Hermann 2008), pp. 199-226.
[6]	J. E. HUMPREYS, Reflection Groups and Coxeter Groups. Cambridge Uni- versity Press, 1990.
[7]	D. REVUZ - M. YOR, Continuous Martingales and Brownian Motion. Spring- er, third edition, 1999.