| Issue |
ESAIM: PS
Volume 21, 2017
|
|
|---|---|---|
| Page(s) | 220 - 234 | |
| DOI | https://doi.org/10.1051/ps/2017006 | |
| Published online | 12 December 2017 | |
Tutte’s invariant approach for Brownian motion reflected in the quadrant
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.
sandro.franceschi@upmc.fr
3 CNRS & Fédération de recherche Denis Poisson & Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France.
Kilian.Raschel@lmpt.univ-tours.fr
Received: 3 June 2016
Revised: 13 December 2016
Accepted: 13 March 2017
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.
Mathematics Subject Classification: 60C05 / 60J65 / 60E10
Key words: Reflected Brownian motion in the quarter plane / stationary distribution / Laplace transform / Tutte’s invariant approach / generalized Chebyshev polynomials
© EDP Sciences, SMAI, 2017
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