Tutte’s invariant approach for Brownian motion reflected in the quadrant

Franceschi, S.; Raschel, Kilian

doi:10.1051/ps/2017006

Franceschi, S.^1,² ; Raschel, Kilian ³

¹ Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris cedex 05, France.
² Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France.
³ CNRS & Fédération de recherche Denis Poisson & Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France.

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 220-234.

Résumé

We consider a Brownian motion with negative drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte’s invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.

Reçu le : 2016-06-03
Accepté le : 2017-03-13

MR Zbl

DOI : 10.1051/ps/2017006

Classification : 60C05, 60J65, 60E10
Mots-clés : Reflected Brownian motion in the quarter plane, stationary distribution, Laplace transform, Tutte’s invariant approach, generalized Chebyshev polynomials

Affiliations des auteurs :

Franceschi, S. ^{1, 2} ; Raschel, Kilian ³

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     author = {Franceschi, S. and Raschel, Kilian},
     title = {Tutte{\textquoteright}s invariant approach for {Brownian} motion reflected in the quadrant},
     journal = {ESAIM: Probability and Statistics},
     pages = {220--234},
     publisher = {EDP-Sciences},
     volume = {21},
     year = {2017},
     doi = {10.1051/ps/2017006},
     mrnumber = {3743912},
     zbl = {1393.60088},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2017006/}
}

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Franceschi, S.; Raschel, Kilian. Tutte’s invariant approach for Brownian motion reflected in the quadrant. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 220-234. doi : 10.1051/ps/2017006. http://www.numdam.org/articles/10.1051/ps/2017006/

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