Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Integration formulas for Brownian motion on classical compact Lie groups

Antoine Dahlqvist

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Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur–Weyl dualities based on stochastic calculus.


On obtient ici des formules combinatoires pour les moments du mouvement Brownien sur les groupes de Lie classiques compacts. Elles sont des déformations de celles obtenues par B. Collins et P. Śniady pour les moments de la mesure de Haar et permettent de donner une preuve fondée sur le calcul stochastique du premier théorème fondamental de la théorie des invariants et de la dualité de Schur–Weyl.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1971-1990.

Received: 3 December 2014
Revised: 7 July 2016
Accepted: 11 July 2016
First available in Project Euclid: 27 November 2017

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Zentralblatt MATH identifier

Primary: 46L53: Noncommutative probability and statistics 60J65: Brownian motion [See also 58J65] 60H30: Applications of stochastic analysis (to PDE, etc.) 60G15: Gaussian processes 14L35: Classical groups (geometric aspects) [See also 20Gxx, 51N30] 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 15A72: Vector and tensor algebra, theory of invariants [See also 13A50, 14L24] 16W22: Actions of groups and semigroups; invariant theory 05E10: Combinatorial aspects of representation theory [See also 20C30]

Brownian motion on Compact Lie groups Weingarten calculus Schur–Weyl duality First Fundamental Theorem of invariant theory


Dahlqvist, Antoine. Integration formulas for Brownian motion on classical compact Lie groups. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1971--1990. doi:10.1214/16-AIHP779.

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