Integral expression for the stationary distribution of reflected Brownian motion in a wedge. (English) Zbl 1428.62081

Franceschi, Sandro; Raschel, Kilian

Bernoulli 25, No. 4B, 3673-3713 (2019).

Summary: 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.

MSC:

60J65	Brownian motion
60K25	Queueing theory (aspects of probability theory)

Keywords:

boundary value problem with shift; Carleman-type boundary value problem; conformal mapping; Laplace transform; reflected Brownian motion in a wedge; stationary distribution

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. · Zbl 0869.60036 · doi:10.1214/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. · Zbl 0634.60085 · doi:10.1137/0147090
[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. · Zbl 1498.05012
[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. · Zbl 1354.05006 · doi:10.1007/s00026-016-0328-7
[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. · Zbl 1209.05008
[6]	Bramson, M. (2011). Positive recurrence for reflecting Brownian motion in higher dimensions. Queueing Syst.69 203-215. · Zbl 1243.60063 · doi:10.1007/s11134-011-9211-8
[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. · Zbl 1200.60066 · doi:10.1214/09-AAP631
[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. · Zbl 1392.60069 · doi:10.1214/16-AOP1130
[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. · Zbl 0860.60062 · doi:10.1214/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. · Zbl 1370.60086 · doi:10.1017/S026996481400031X
[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. · Zbl 1348.60068 · doi:10.1007/s11134-016-9483-0
[12]	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.
[13]	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. · Zbl 0786.60107 · doi:10.1214/aoap/1177005771
[14]	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. · Zbl 1260.60165 · doi:10.1007/s00186-010-0304-7
[15]	Dai, J.G. and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stoch. Syst.1 146-208. · Zbl 1291.60168 · doi:10.1287/10-SSY022
[16]	Dai, J.G. and Miyazawa, M. (2013). Stationary distribution of a two-dimensional SRBM: Geometric views and boundary measures. Queueing Syst.74 181-217. · Zbl 1276.60089 · doi:10.1007/s11134-012-9339-1
[17]	Dieker, A.B. and Moriarty, J. (2009). Reflected Brownian motion in a wedge: Sum-of-exponential stationary densities. Electron. Commun. Probab.14 1-16. · Zbl 1190.60077 · doi:10.1214/ECP.v14-1437
[18]	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. · Zbl 0278.44001
[19]	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. · Zbl 1392.05007 · doi:10.1007/s00222-018-0787-z
[20]	Dubédat, J. (2004). Reflected planar Brownian motions, intertwining relations and crossing probabilities. Ann. Inst. Henri Poincaré Probab. Stat.40 539-552. · Zbl 1054.60085
[21]	Dupuis, P. and Williams, R.J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab.22 680-702. · Zbl 0808.60068 · doi:10.1214/aop/1176988725
[22]	Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: The reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete47 325-351. · Zbl 0395.68032 · doi:10.1007/BF00535168
[23]	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. · Zbl 0932.60002
[24]	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.
[25]	Foschini, G.J. (1982). Equilibria for diffusion models of pairs of communicating computers—symmetric case. IEEE Trans. Inform. Theory28 273-284. · Zbl 0476.68030 · doi:10.1109/TIT.1982.1056473
[26]	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. · Zbl 1365.60072 · doi:10.1287/16-SSY218
[27]	Franceschi, S. and Raschel, K. (2017). Tutte’s invariant approach for Brownian motion reflected in the quadrant. ESAIM Probab. Stat.21 220-234. · Zbl 1393.60088 · doi:10.1051/ps/2017006
[28]	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. · Zbl 1166.60331 · doi:10.1007/s11134-008-9102-9
[29]	Harrison, J.M. and Reiman, M.I. (1981). On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math.41 345-361. · Zbl 0464.60081 · doi:10.1137/0141030
[30]	Harrison, J.M. and Reiman, M.I. (1981). Reflected Brownian motion on an orthant. Ann. Probab.9 302-308. · Zbl 0462.60073 · doi:10.1214/aop/1176994471
[31]	Harrison, J.M. and Williams, R.J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics22 77-115. · Zbl 0632.60095 · doi:10.1080/17442508708833469
[32]	Harrison, J.M. and Williams, R.J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab.15 115-137. · Zbl 0615.60072 · doi:10.1214/aop/1176992259
[33]	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. · Zbl 0776.60100 · doi:10.1017/S0305004100076040
[34]	Jackson, J.R. (1957). Networks of waiting lines. Oper. Res.5 518-521. · Zbl 1414.90067 · doi:10.1287/opre.5.4.518
[35]	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. · Zbl 1248.90029 · doi:10.1287/moor.1120.0538
[36]	Kella, O. and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Probab.33 1169-1180. · Zbl 0867.60081 · doi:10.2307/3214994
[37]	Kurkova, I.A. and Suhov, Y.M. (2003). Malyshev’s theory and JS-queues. Asymptotics of stationary probabilities. Ann. Appl. Probab.13 1313-1354. · Zbl 1039.60082 · doi:10.1214/aoap/1069786501
[38]	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.
[39]	Latouche, G. and Miyazawa, M. (2014). Product-form characterization for a two-dimensional reflecting random walk. Queueing Syst.77 373-391. · Zbl 1310.60050 · doi:10.1007/s11134-013-9381-7
[40]	Le Gall, J.-F. (1987). Mouvement brownien, cônes et processus stables. Probab. Theory Related Fields76 587-627. · Zbl 0611.60076 · doi:10.1007/BF00960076
[41]	Lépingle, D. (2017). A two-dimensional oblique extension of Bessel processes. Markov Process. Related Fields23 233-266. · Zbl 1379.60060
[42]	Litvinchuk, G.S. (2000). Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and Its Applications523. Dordrecht: Kluwer Academic.
[43]	Malyšev, V.A. (1972). An analytic method in the theory of two-dimensional positive random walks. Sibirsk. Mat. Zh.13 1314-1329, 1421.
[44]	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.
[45]	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. · Zbl 1295.60094
[46]	Reiman, M.I. and Williams, R.J. (1988). A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields77 87-97. · Zbl 0617.60081 · doi:10.1007/BF01848132
[47]	Sarantsev, A. (2013). Comparison techniques for competing brownian particles. 1-31. Preprint. Available at arXiv:1305.1653. · Zbl 1478.60270
[48]	Sarantsev, A. (2015). Triple and simultaneous collisions of competing Brownian particles. Electron. J. Probab.20 no. 29, 28. · Zbl 1321.60213 · doi:10.1214/EJP.v20-3279
[49]	Sarantsev, A. (2017). Reflected Brownian motion in a convex polyhedral cone: Tail estimates for the stationary distribution. J. Theoret. Probab.30 1200-1223. · Zbl 1378.60105 · doi:10.1007/s10959-016-0674-8
[50]	Takayama, N. (1992). An approach to the zero recognition problem by Buchberger algorithm. J. Symbolic Comput.14 265-282. · Zbl 0763.65007 · doi:10.1016/0747-7171(92)90039-7
[51]	Taylor, L.M. and Williams, R.J. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields96 283-317. · Zbl 0794.60079 · doi:10.1007/BF01292674
[52]	Tutte, W.T. (1995). Chromatic sums revisited. Aequationes Math.50 95-134. · Zbl 0842.05031 · doi:10.1007/BF01831115
[53]	Varadhan, S.R.S. and Williams, R.J. (1985). Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math.38 405-443. · Zbl 0579.60082 · doi:10.1002/cpa.3160380405
[54]	Williams, R.J. (1985). Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Probab.13 758-778. · Zbl 0596.60078 · doi:10.1214/aop/1176992907
[55]	Williams, R.J. (1985). Reflected Brownian motion in a wedge: Semimartingale property. Z. Wahrsch. Verw. Gebiete69 161-176. · Zbl 0535.60042 · doi:10.1007/BF02450279
[56]	Williams, R.J. (1995). Semimartingale reflecting Brownian motions in the orthant. In Stochastic Networks. IMA Vol. Math. Appl.71 125-137. New York: Springer. · Zbl 0827.60031