This survey establishes some miscellaneous results on random Littelmann paths and generalized Pitman transform. We describe central probability distributions on Littelmann paths. Next we state a law of large numbers and a central limit theorem for the generalized Pitman transform. We then study harmonic functions on multiplicative graphs defined from the tensor powers of finite-dimensional Lie algebras representations. Finally, we explain there exists an inverse of the generalized Pitman transform defined almost surely on the set of infinite paths remaining in the Weyl chamber and how it can be computed.
Dans cet article de synthèse nous établissons des résultats complémentaires sur les chemins de Littelmann aléatoires et sur la transformée de Pitman généralisée. Nous décrivons les distributions de probabilité centrales sur les chemins de Littelmann. Ensuite nous donnons une loi des grands nombres et un théorème central limite pour la transformée de Pitman généralisée. Nous étudions alors les fonctions harmoniques sur les graphes multiplicatifs définis à partir des puissances tensorielles des représentations irréductibles des algèbres de Lie. Enfin, nous expliquons qu’il existe une transformée inverse de la transformée de Pitman généralisée définie presque sûrement sur les trajectoires infinies qui restent dans la chambre de Weyl et montrons comment elle peut être calculée.
Accepted:
Published online:
DOI: 10.5802/afst.1580
Cédric Lecouvey 1; Emmanuel Lesigne 1; Marc Peigné 1
CC-BY 4.0
@article{AFST_2018_6_27_3_629_0,
author = {C\'edric Lecouvey and Emmanuel Lesigne and Marc Peign\'e},
title = {Harmonic functions on multiplicative graphs and inverse {Pitman} transform on infinite random paths},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {629--666},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 27},
number = {3},
year = {2018},
doi = {10.5802/afst.1580},
zbl = {1400.05262},
mrnumber = {3869076},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1580/}
}
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%0 Journal Article %A Cédric Lecouvey %A Emmanuel Lesigne %A Marc Peigné %T Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2018 %P 629-666 %V 27 %N 3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1580/ %R 10.5802/afst.1580 %G en %F AFST_2018_6_27_3_629_0
Cédric Lecouvey; Emmanuel Lesigne; Marc Peigné. Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 3, pp. 629-666. doi : 10.5802/afst.1580. https://afst.centre-mersenne.org/articles/10.5802/afst.1580/
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