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

@article{PS_2017__21__220_0,
     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://archive.numdam.org/articles/10.1051/ps/2017006/}
}

TY  - JOUR
AU  - Franceschi, S.
AU  - Raschel, Kilian
TI  - Tutte’s invariant approach for Brownian motion reflected in the quadrant
JO  - ESAIM: Probability and Statistics
PY  - 2017
SP  - 220
EP  - 234
VL  - 21
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2017006/
DO  - 10.1051/ps/2017006
LA  - en
ID  - PS_2017__21__220_0
ER  -

%0 Journal Article
%A Franceschi, S.
%A Raschel, Kilian
%T Tutte’s invariant approach for Brownian motion reflected in the quadrant
%J ESAIM: Probability and Statistics
%D 2017
%P 220-234
%V 21
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2017006/
%R 10.1051/ps/2017006
%G en
%F PS_2017__21__220_0

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://archive.numdam.org/articles/10.1051/ps/2017006/

Bibliographie
Cité par

F. Baccelli and G. Fayolle, Analysis of models reducible to a class of diffusion processes in the positive quarter plane. SIAM J. Appl. Math. 47 (1987) 1367–1385. | DOI | MR | Zbl

O. Bernardi, M. Bousquet-Mélou and K. Raschel, Counting quadrant walks via Tutte’s invariant method. In 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2016) 203–214.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane. In Algorithmic probability and combinatorics. Vol. 520 of Contemp. Math. Amer. Math. Soc. Providence, RI (2010) 1–39. | MR | Zbl

K. Burdzy, Z.-Q. Chen, D. Marshall and K. Ramanan, Obliquely reflected Brownian motion in non-smooth planar domains. Ann. Probab. 45 (2017) 2971–3037. | DOI | MR | Zbl

J. Dai, Steady-state analysis of reflected Brownian motions: Characterization, numerical methods and queueing applications. ProQuest LLC, Ann Arbor, MI. Ph.D. thesis, Stanford University (1990). | MR

J. Dai and A. Dieker, Nonnegativity of solutions to the basic adjoint relationship for some diffusion processes. Queueing Syst. 68 (2011) 295–303. | DOI | MR | Zbl

J. Dai, S. Guettes and T. Kurtz, Characterization of the stationary distribution for a reflecting brownian motion in a convex polyhedron. Tech. Rep., Department of Mathematics, University of Wisconsin-Madison (2010).

J. Dai and J. Harrison, Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann. Appl. Probab. 2 (1992) 65–86. | MR | Zbl

J. Dai and T. Kurtz, Characterization of the stationary distribution for a semimartingale reflecting brownian motion in a convex polyhedron (1994).

J. Dai and M. Miyazawa, Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stoch. Syst. 1 (2011)146–208. | MR | Zbl

J. Dai and M. Miyazawa, Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures. Queueing Syst. 74 (2013) 181–217. | DOI | MR | Zbl

A. Dieker and J. Moriarty, Reflected Brownian motion in a wedge: sum-of-exponential stationary densities. Electron. Commun. Probab.14 (2009) 1–16. | DOI | MR | Zbl

G. Doetsch, Introduction to the theory and application of the Laplace transformation. Springer-Verlag, New York Heidelberg (1974). | MR | Zbl

J. Dubédat, Reflected planar Brownian motions, intertwining relations and crossing probabilities. Ann. Inst. Henri Poincaré Probab. Statist. 40 (2004) 539–552. | DOI | Numdam | MR | Zbl

P. Dupuis and R. Williams, Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 (1994) 680–702. | DOI | MR | Zbl

G. Fayolle, R. Iasnogorodski and V. Malyshev, Random Walks in the Quarter-Plane. Springer Berlin Heidelberg, Berlin, Heidelberg (1999). | MR | Zbl

M. Foddy, Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem). ProQuest LLC, Ann Arbor, MI. Ph.D. thesis, Stanford University (1984). | MR

G. Foschini, Equilibria for diffusion models of pairs of communicating computers–symmetric case. IEEE Trans. Inform. Theory 28 (1982) 273–284. | DOI | MR | Zbl

S. Franceschi and I. Kurkova, Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane. Preprint (2016). | arXiv | MR

S. Franceschi, I. Kurkova and K. Raschel, Analytic approach for reflected Brownian motion in the quadrant. In 27th Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA’16) (2016). | MR

J. Harrison and J. Hasenbein, Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution. Queueing Syst. 61 (2009) 113–138. | DOI | MR | Zbl

J. Harrison and M. Reiman, On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math. 41 (1981) 345–361. | DOI | MR | Zbl

J. Harrison and R. Williams, Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22 (1987a) 77–115. | DOI | MR | Zbl

J. Harrison and R. Williams, Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 (1987b) 115–137. | DOI | MR | Zbl

D. Hobson and L. Rogers, Recurrence and transience of reflecting Brownian motion in the quadrant. In vol. 113 of Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge Univ Press (1993) 387–399 | MR | Zbl

J.-F. Le Gall, Mouvement brownien, cônes et processus stables. Probab. Theory Related Fields 76 (1987) 587–627. | DOI | MR | Zbl

G. Litvinchuk, Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Springer Netherlands, Dordrecht (2000). | MR | Zbl

V. Malyšev, An analytic method in the theory of two-dimensional positive random walks. Sibirsk. Mat. Ž. 13 (1972) 1314–1329, 1421. | MR | Zbl

W. Tutte, Chromatic sums revisited. Aequationes Math. 50 (1995) 95–134. | DOI | MR | Zbl

R. Williams, Semimartingale reflecting Brownian motions in the orthant. In Stochastic networks. Vol. 71 of IMA Vol. Math. Appl. Springer, New York (1995) 125–137. | MR | Zbl

Cité par Sources :