The Annals of Applied Probability

Multidimensional sticky Brownian motions as limits of exclusion processes

Miklós Z. Rácz and Mykhaylo Shkolnikov

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We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and the entire particle system is slowed down until the “collision” is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and reflects at the boundary of the wedge after spending an instant of time there. In particular, this leads to a natural multidimensional generalization of sticky Brownian motion on the half-line, which is of interest in both queuing theory and stochastic portfolio theory. For instance, this can model a market, which experiences a slowdown due to a major event (such as a court trial between some of the largest firms in the market) deciding about the new market leader.

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Ann. Appl. Probab. Volume 25, Number 3 (2015), 1155-1188.

First available in Project Euclid: 23 March 2015

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Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60H10: Stochastic ordinary differential equations [See also 34F05]

Exclusion processes with speed change reflected Brownian motion scaling limits sticky Brownian motion stochastic differential equations stochastic portfolio theory


Rácz, Miklós Z.; Shkolnikov, Mykhaylo. Multidimensional sticky Brownian motions as limits of exclusion processes. Ann. Appl. Probab. 25 (2015), no. 3, 1155--1188. doi:10.1214/14-AAP1019.

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  • [1] Amir, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. Stochastic Process. Appl. 39 221–237.
  • [2] Bass, R. F. and Pardoux, É. (1987). Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 557–572.
  • [3] Bhardwaj, S. and Williams, R. J. (2009). Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation. Queueing Syst. 62 345–382.
  • [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [5] Cherny, A. S. (2002). On the uniqueness in law and the pathwise uniqueness for stochastic differential equations. Theory Probab. Appl. 46 406–419.
  • [6] Chitashvili, R. J. (1989). On the nonexistence of a strong solution in the boundary problem for a sticky Brownian motion. Centrum voor Wiskunde en Informatica, Centre for Mathematics and Computer Science, Report BS-R8901.
  • [7] Dupuis, P. and Williams, R. J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 680–702.
  • [8] Émery, M. and Schachermayer, W. (2001). On Vershik’s standardness criterion and Tsirelson’s notion of cosiness. In Séminaire de Probabilités. XXXV. Lecture Notes in Math. 1755 265–305. Springer, Berlin.
  • [9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [10] Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 468–519.
  • [11] Feller, W. (1954). Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77 1–31.
  • [12] Harrison, J. M. and Lemoine, A. J. (1981). Sticky Brownian motion as the limit of storage processes. J. Appl. Probab. 18 216–226.
  • [13] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.
  • [14] Harrison, J. M. and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115–137.
  • [15] Ichiba, T. (2009). Topics in multidimensional diffusion theory: Attainability, reflection, ergodicity and rankings. PhD dissertation, Columbia Univ., New York.
  • [16] Ichiba, T. and Karatzas, I. (2010). On collisions of Brownian particles. Ann. Appl. Probab. 20 951–977.
  • [17] Itô, K. and McKean, H. P. Jr. (1963). Brownian motions on a half line. Illinois J. Math. 7 181–231.
  • [18] Itô, K. and McKean, H. P. Jr. (1996). Diffusion Processes and Their Sample Paths. Springer, Berlin. Reprint of the 1974 edition.
  • [19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [20] Karatzas, I., Pal, S. and Shkolnikov, M. (2012). Systems of Brownian particles with asymmetric collisions. Preprint. Available at arXiv:1210.0259.
  • [21] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [22] Lemoine, A. J. (1974). Limit theorems for generalized single server queues. Adv. in Appl. Probab. 6 159–174.
  • [23] Lemoine, A. J. (1975). Limit theorems for generalized single server queues: The exceptional system. SIAM J. Appl. Math. 28 596–606.
  • [24] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. in Appl. Probab. 25 487–517.
  • [25] Reiman, M. I. and Williams, R. J. (1988). A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 87–97.
  • [26] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [27] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin. Reprint of the 1997 edition.
  • [28] 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.
  • [29] Warren, J. (1997). Branching processes, the Ray–Knight theorem, and sticky Brownian motion. In Séminaire de Probabilités, XXXI. 1–15. Springer, Berlin.
  • [30] Welch, P. D. (1964). On a generalized $M/G/1$ queuing process in which the first customer of each busy period receives exceptional service. Oper. Res. 12 736–752.
  • [31] Williams, R. J. (1995). Semimartingale reflecting Brownian motions in the orthant. In Stochastic Networks. IMA Vol. Math. Appl. 71 125–137. Springer, New York.
  • [32] Williams, R. J. (1998). An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Syst. Theory Appl. 30 5–25.
  • [33] Yeo, G. F. (1961/1962). Single server queues with modified service mechanisms. J. Aust. Math. Soc. 2 499–507.