The Annals of Applied Probability

On collisions of Brownian particles

Tomoyuki Ichiba and Ioannis Karatzas

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We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence and for the presence of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains.

Article information

Ann. Appl. Probab. Volume 20, Number 3 (2010), 951-977.

First available in Project Euclid: 18 June 2010

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Zentralblatt MATH identifier

Primary: 60G17: Sample path properties 60G44: Martingales with continuous parameter
Secondary: 60G85

Martingale problem triple collision effective dimension Bessel process reflected Brownian motion comparison theorem Atlas model


Ichiba, Tomoyuki; Karatzas, Ioannis. On collisions of Brownian particles. Ann. Appl. Probab. 20 (2010), no. 3, 951--977. doi:10.1214/09-AAP641.

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  • [1] Banner, A. D., Fernholz, R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296–2330.
  • [2] Banner, A. D. and Ghomrasni, R. (2008). Local times of ranked continuous semimartingales. Stochastic Process. Appl. 118 1244–1253.
  • [3] Bass, R. F. and Pardoux, É. (1987). Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 557–572.
  • [4] Burdzy, K. and Marshall, D. (1992). Hitting a boundary point with reflected Brownian motion. In Séminaire de Probabilités, XXVI. Lecture Notes in Mathematics 1526 81–94. Springer, Berlin.
  • [5] Cépa, E. and Lépingle, D. (2007). No multiple collisions for mutually repelling Brownian particles. In Séminaire de Probabilités XL. Lecture Notes in Mathematics 1899 241–246. Springer, Berlin.
  • [6] Delarue, F. (2008). Hitting time of a corner for a reflected diffusion in the square. Ann. Inst. Henri Poincaré Probab. Stat. 44 946–961.
  • [7] Elworthy, K. D., Li, X.-M. and Yor, M. (1999). The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes. Probab. Theory Related Fields 115 325–355.
  • [8] Fernholz, E. R. (2002). Stochastic Portfolio Theory. Stochastic Modelling and Applied Probability 48. Springer, New York.
  • [9] Friedman, A. (1974). Nonattainability of a set by a diffusion process. Trans. Amer. Math. Soc. 197 245–271.
  • [10] Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9 313–349.
  • [11] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.
  • [12] Harrison, J. M. and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22 77–115.
  • [13] Ichiba, T. (2006). Note on Atlas model of equity markets. Mimeo.
  • [14] Ikeda, N. and Watanabe, S. (1977). A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14 619–633.
  • [15] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [16] Krylov, N. V. (1980). Controlled Diffusion Processes. Applications of Mathematics 14. Springer, New York.
  • [17] Krylov, N. V. (2004). On weak uniqueness for some diffusions with discontinuous coefficients. Stochastic Process. Appl. 113 37–64.
  • [18] Meyers, N. and Serrin, J. (1960). The exterior Dirichlet problem for second order elliptic partial differential equations. J. Math. Mech. 9 513–538.
  • [19] Pal, S. and Pitman, J. (2008). One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18 2179–2207.
  • [20] Ramasubramanian, S. (1983). Recurrence of projections of diffusions. Sankhyā Ser. A 45 20–31.
  • [21] Ramasubramanian, S. (1988). Hitting of submanifolds by diffusions. Probab. Theory Related Fields 78 149–163.
  • [22] Rogers, L. C. G. (1990). Brownian motion in a wedge with variable skew reflection. II. In Diffusion Processes and Related Problems in Analysis I. Progress in Probability 22 95–115. Birkhäuser, Boston, MA.
  • [23] Rogers, L. C. G. (1991). Brownian motion in a wedge with variable skew reflection. Trans. Amer. Math. Soc. 326 227–236.
  • [24] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin.
  • [25] Varadhan, S. R. S. and Williams, R. J. (1985). Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 405–443.
  • [26] Williams, R. J. (1987). Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 459–485.