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Comptes Rendus Mathématique
Volume 347, n° 19-20
pages 1125-1128 (octobre 2009)
Doi : 10.1016/j.crma.2009.08.003
Received : 12 May 2009 ;  accepted : 20 August 2009
Radial Dunkl processes: Existence, uniqueness and hitting time
Processus de Dunkl radial : Existence, unicité et temps d’atteinte
 

Nizar Demni
SFB, Bielefeld University, Universität Strasse, 33501 Bielefeld, Germany 

Abstract

We give shorter proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all times t of a stochastic differential equation with a singular drift, the first hitting time of the Weyl chamber by a radial Dunkl process is finite almost surely for small values of the multiplicity function. The proof of the first result allows one to give a positive answer to a conjecture announced by Gallardo–Yor while that of the second shows that the process hits almost surely the wall corresponding to the simple root with a small multiplicity value. To cite this article: N. Demni, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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Résumé

On donne de courtes preuves des résultats suivants : le processus de Dunkl radial associé à un système de racines réduit et une fonction de multiplicité strictement positive est l’unique solution forte d’une équation différentielle stochastique à dérive singulière pour tout temps t , le temps d’atteinte de la frontière de la chambre de Weyl est fini presque sûrement pour les petites valeurs de la fonction de multiplicité. La preuve du premier résultat permet de donner une réponse positive à une conjecture de Gallardo–Yor, alors que celle du deuxième résultat montre que le processus touche précisemment le mur correspondant à la racine simple pour laquelle la valeur de la multiplicité est suffisamment petite. Pour citer cet article : N. Demni, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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