Bernoulli
- Bernoulli
- Volume 25, Number 4B (2019), 3673-3713.
Integral expression for the stationary distribution of reflected Brownian motion in a wedge
Sandro Franceschi and Kilian Raschel
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Abstract
For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polynomials. To that purpose, we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.
Article information
Source
Bernoulli, Volume 25, Number 4B (2019), 3673-3713.
Dates
Received: January 2018
Revised: September 2018
First available in Project Euclid: 25 September 2019
Permanent link to this document
https://projecteuclid.org/euclid.bj/1569398781
Digital Object Identifier
doi:10.3150/19-BEJ1107
Mathematical Reviews number (MathSciNet)
MR4010969
Zentralblatt MATH identifier
07110152
Keywords
boundary value problem with shift Carleman-type boundary value problem conformal mapping Laplace transform reflected Brownian motion in a wedge stationary distribution
Citation
Franceschi, Sandro; Raschel, Kilian. Integral expression for the stationary distribution of reflected Brownian motion in a wedge. Bernoulli 25 (2019), no. 4B, 3673--3713. doi:10.3150/19-BEJ1107. https://projecteuclid.org/euclid.bj/1569398781
References
- [1] Aspandiiarov, S., Iasnogorodski, R. and Menshikov, M. (1996). Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant. Ann. Probab. 24 932–960.Zentralblatt MATH: 0869.60036
Digital Object Identifier: doi:10.1214/aop/1039639371
Project Euclid: euclid.aop/1039639371 - [2] Baccelli, F. and Fayolle, G. (1987). Analysis of models reducible to a class of diffusion processes in the positive quarter plane. SIAM J. Appl. Math. 47 1367–1385.
- [3] Bernardi, O., Bousquet-Mélou, M. and Raschel, K. (2016). Counting quadrant walks via Tutte’s invariant method. In Proceedings of FPSAC 2016. Discrete Math. Theor. Comput. Sci. Proc. 203–214. Nancy: Assoc. Discrete Math. Theor. Comput. Sci.
- [4] Bostan, A., Bousquet-Mélou, M., Kauers, M. and Melczer, S. (2016). On 3-dimensional lattice walks confined to the positive octant. Ann. Comb. 20 661–704.
- [5] Bousquet-Mélou, M. and Mishna, M. (2010). Walks with small steps in the quarter plane. In Algorithmic Probability and Combinatorics. Contemp. Math. 520 1–39. Providence, RI: Amer. Math. Soc.Zentralblatt MATH: 1209.05008
- [6] Bramson, M. (2011). Positive recurrence for reflecting Brownian motion in higher dimensions. Queueing Syst. 69 203–215.
- [7] Bramson, M., Dai, J.G. and Harrison, J.M. (2010). Positive recurrence of reflecting Brownian motion in three dimensions. Ann. Appl. Probab. 20 753–783.Zentralblatt MATH: 1200.60066
Digital Object Identifier: doi:10.1214/09-AAP631
Project Euclid: euclid.aoap/1268143439 - [8] Burdzy, K., Chen, Z.-Q., Marshall, D. and Ramanan, K. (2017). Obliquely reflected Brownian motion in nonsmooth planar domains. Ann. Probab. 45 2971–3037.Zentralblatt MATH: 1392.60069
Digital Object Identifier: doi:10.1214/16-AOP1130
Project Euclid: euclid.aop/1506132032 - [9] Chen, H. (1996). A sufficient condition for the positive recurrence of a semimartingale reflecting Brownian motion in an orthant. Ann. Appl. Probab. 6 758–765.Zentralblatt MATH: 0860.60062
Digital Object Identifier: doi:10.1214/aoap/1034968226
Project Euclid: euclid.aoap/1034968226 - [10] Chen, Y., Boucherie, R.J. and Goseling, J. (2015). The invariant measure of random walks in the quarter-plane: Representation in geometric terms. Probab. Engrg. Inform. Sci. 29 233–251.
- [11] Chen, Y., Boucherie, R.J. and Goseling, J. (2016). Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms. Queueing Syst. 84 21–48.
- [12] Cohen, J. (1984). On a functional relation in three complex variables; three coupled processors Technical Report Mathematical, Institute Utrecht 359, Utrecht University.
- [13] Dai, J. (1990). Steady-State Analysis of Reflected Brownian Motions: Characterization, Numerical Methods and Queueing Applications. Ann Arbor, MI: ProQuest LLC. Thesis (Ph.D.)–Stanford University.
- [14] Dai, J. and Kurtz, T. (1994). Characterization of the stationary distribution for a semimartingale reflecting brownian motion in a convex polyhedron. 1–31. Preprint.
- [15] Dai, J.G. and Harrison, J.M. (1992). Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Ann. Appl. Probab. 2 65–86.Zentralblatt MATH: 0786.60107
Digital Object Identifier: doi:10.1214/aoap/1177005771
Project Euclid: euclid.aoap/1177005771 - [16] Dai, J.G. and Harrison, J.M. (2012). Reflecting Brownian motion in three dimensions: A new proof of sufficient conditions for positive recurrence. Math. Methods Oper. Res. 75 135–147.
- [17] Dai, J.G. and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stoch. Syst. 1 146–208.
- [18] Dai, J.G. and Miyazawa, M. (2013). Stationary distribution of a two-dimensional SRBM: Geometric views and boundary measures. Queueing Syst. 74 181–217.
- [19] Dieker, A.B. and Moriarty, J. (2009). Reflected Brownian motion in a wedge: Sum-of-exponential stationary densities. Electron. Commun. Probab. 14 1–16.
- [20] Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transformation. New York: Springer. Translated from the second German edition by Walter Nader.Zentralblatt MATH: 0278.44001
- [21] Dreyfus, T., Hardouin, C., Roques, J. and Singer, M.F. (2018). On the nature of the generating series of walks in the quarter plane. Invent. Math. 213 139–203.
- [22] Dubédat, J. (2004). Reflected planar Brownian motions, intertwining relations and crossing probabilities. Ann. Inst. Henri Poincaré Probab. Stat. 40 539–552.
- [23] Dupuis, P. and Williams, R.J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 680–702.Zentralblatt MATH: 0808.60068
Digital Object Identifier: doi:10.1214/aop/1176988725
Project Euclid: euclid.aop/1176988725 - [24] Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: The reduction to a Riemann–Hilbert problem. Z. Wahrsch. Verw. Gebiete 47 325–351.
- [25] Fayolle, G., Iasnogorodski, R. and Malyshev, V. (1999). Random Walks in the Quarter-Plane. Algebraic Methods, Boundary Value Problems and Applications. Applications of Mathematics (New York) 40. Berlin: Springer.Zentralblatt MATH: 0932.60002
- [26] Foddy, M.E. (1984). Analysis of Brownian Motion with Drift, Confined to a Quadrant by Oblique Reflection (Diffusions, Riemann–Hilbert Problem). Ann Arbor, MI: ProQuest LLC. Thesis (Ph.D.)–Stanford University.
- [27] Foschini, G.J. (1982). Equilibria for diffusion models of pairs of communicating computers—symmetric case. IEEE Trans. Inform. Theory 28 273–284.
- [28] Franceschi, S. and Kourkova, I. (2017). Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach. Stoch. Syst. 7 32–94.
- [29] Franceschi, S. and Raschel, K. (2017). Tutte’s invariant approach for Brownian motion reflected in the quadrant. ESAIM Probab. Stat. 21 220–234.
- [30] Harrison, J.M. and Hasenbein, J.J. (2009). Reflected Brownian motion in the quadrant: Tail behavior of the stationary distribution. Queueing Syst. 61 113–138.
- [31] Harrison, J.M. and Reiman, M.I. (1981). On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math. 41 345–361.
- [32] Harrison, J.M. and Reiman, M.I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.Zentralblatt MATH: 0462.60073
Digital Object Identifier: doi:10.1214/aop/1176994471
Project Euclid: euclid.aop/1176994471 - [33] Harrison, J.M. and Williams, R.J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22 77–115.
- [34] Harrison, J.M. and Williams, R.J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115–137.Zentralblatt MATH: 0615.60072
Digital Object Identifier: doi:10.1214/aop/1176992259
Project Euclid: euclid.aop/1176992259 - [35] Hobson, D.G. and Rogers, L.C.G. (1993). Recurrence and transience of reflecting Brownian motion in the quadrant. Math. Proc. Cambridge Philos. Soc. 113 387–399.
- [36] Jackson, J.R. (1957). Networks of waiting lines. Oper. Res. 5 518–521.
- [37] Kella, O. and Ramasubramanian, S. (2012). Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant. Math. Oper. Res. 37 301–312.Mathematical Reviews (MathSciNet): MR2931282
Zentralblatt MATH: 1248.90029
Digital Object Identifier: doi:10.1287/moor.1120.0538 - [38] Kella, O. and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Probab. 33 1169–1180.
- [39] Kurkova, I.A. and Suhov, Y.M. (2003). Malyshev’s theory and JS-queues. Asymptotics of stationary probabilities. Ann. Appl. Probab. 13 1313–1354.Zentralblatt MATH: 1039.60082
Digital Object Identifier: doi:10.1214/aoap/1069786501
Project Euclid: euclid.aoap/1069786501 - [40] Lakner, P., Reed, J. and Zwart, B. (2016). A Dirichlet process characterization of RBM in a wedge. 1–37. Preprint. Available at arXiv:1605.02020.arXiv: 1605.02020
- [41] Latouche, G. and Miyazawa, M. (2014). Product-form characterization for a two-dimensional reflecting random walk. Queueing Syst. 77 373–391.
- [42] Le Gall, J.-F. (1987). Mouvement brownien, cônes et processus stables. Probab. Theory Related Fields 76 587–627.
- [43] Lépingle, D. (2017). A two-dimensional oblique extension of Bessel processes. Markov Process. Related Fields 23 233–266.Zentralblatt MATH: 1379.60060
- [44] Litvinchuk, G.S. (2000). Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and Its Applications 523. Dordrecht: Kluwer Academic.
- [45] Malyšev, V.A. (1972). An analytic method in the theory of two-dimensional positive random walks. Sibirsk. Mat. Zh. 13 1314–1329, 1421.Mathematical Reviews (MathSciNet): MR0336823
- [46] Muskhelishvili, N.I. (1972). Singular Integral Equations. Boundary Problems of Functions Theory and Their Applications to Mathematical Physics. Groningen: Wolters-Noordhoff Publishing. Revised translation from the Russian, edited by J. R. M. Radok, Reprinted.
- [47] O’Connell, N. and Ortmann, J. (2014). Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition. ALEA Lat. Am. J. Probab. Math. Stat. 11 307–329.Zentralblatt MATH: 1295.60094
- [48] Reiman, M.I. and Williams, R.J. (1988). A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 87–97.
- [49] Sarantsev, A. (2013). Comparison techniques for competing brownian particles. 1–31. Preprint. Available at arXiv:1305.1653.arXiv: 1305.1653
- [50] Sarantsev, A. (2015). Triple and simultaneous collisions of competing Brownian particles. Electron. J. Probab. 20 no. 29, 28.
- [51] Sarantsev, A. (2017). Reflected Brownian motion in a convex polyhedral cone: Tail estimates for the stationary distribution. J. Theoret. Probab. 30 1200–1223.
- [52] Takayama, N. (1992). An approach to the zero recognition problem by Buchberger algorithm. J. Symbolic Comput. 14 265–282.
- [53] Taylor, L.M. and Williams, R.J. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 96 283–317.
- [54] Tutte, W.T. (1995). Chromatic sums revisited. Aequationes Math. 50 95–134.
- [55] Varadhan, S.R.S. and Williams, R.J. (1985). Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 405–443.
- [56] Williams, R.J. (1985). Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Probab. 13 758–778.Zentralblatt MATH: 0596.60078
Digital Object Identifier: doi:10.1214/aop/1176992907
Project Euclid: euclid.aop/1176992907 - [57] Williams, R.J. (1985). Reflected Brownian motion in a wedge: Semimartingale property. Z. Wahrsch. Verw. Gebiete 69 161–176.
- [58] Williams, R.J. (1995). Semimartingale reflecting Brownian motions in the orthant. In Stochastic Networks. IMA Vol. Math. Appl. 71 125–137. New York: Springer.Zentralblatt MATH: 0827.60031

- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Obliquely reflected Brownian motion in nonsmooth planar domains
Burdzy, Krzysztof, Chen, Zhen-Qing, Marshall, Donald, and Ramanan, Kavita, Annals of Probability, 2017 - Some positive eigenfunctions for elliptic operators with oblique derivative boundary conditions and consequences for the stationary densities of reflected Brownian motions
Williams, Ruth J., , 1986 - On the Distribution of the Integral of the Absolute Value of the Brownian Motion
Takacs, Lajos, Annals of Applied Probability, 1993
- Obliquely reflected Brownian motion in nonsmooth planar domains
Burdzy, Krzysztof, Chen, Zhen-Qing, Marshall, Donald, and Ramanan, Kavita, Annals of Probability, 2017 - Some positive eigenfunctions for elliptic operators with oblique derivative boundary conditions and consequences for the stationary densities of reflected Brownian motions
Williams, Ruth J., , 1986 - On the Distribution of the Integral of the Absolute Value of the Brownian Motion
Takacs, Lajos, Annals of Applied Probability, 1993 - The class of distributions associated with the generalized Pollaczek-Khinchine formula
Kella, Offer, Journal of Applied Probability, 2012 - Reflected Brownian motion in a wedge: sum-of-exponential stationary densities
Dieker, A.B. and Moriarty, J., Electronic Communications in Probability, 2009 - Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas
Janson, Svante, Probability Surveys, 2007 - Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach
Franceschi, Sandro and Kourkova, Irina, Stochastic Systems, 2017 - First passage times of constant-elasticity-of-variance processes
with two-sided reflecting barriers
Bo, Lijun and Hao, Chen, Journal of Applied Probability, 2012 - On the Local Time of the Brownian Motion
Takacs, Lajos, Annals of Applied Probability, 1995 - Double-barrier Parisian options
Dassios, Angelos and Wu, Shanle, Journal of Applied Probability, 2011