Annals of Applied Probability

Concentration of measure for Brownian particle systems interacting through their ranks

Soumik Pal and Mykhaylo Shkolnikov

Full-text: Open access


We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using transportation cost inequalities for stochastic processes we provide uniform fluctuation bounds for the ordered particles, their local time of collisions and various associated statistics over intervals of time. For example, such processes, when exponentiated and rescaled, exhibit power law decay under stationarity; we derive concentration bounds for the empirical estimates of the index of the power law over large intervals of time. A key ingredient in our proofs is a novel upper bound on the Lipschitz constant of the Skorokhod map that transforms a multidimensional Brownian path to a path which is constrained not to leave the positive orthant.

Article information

Ann. Appl. Probab., Volume 24, Number 4 (2014), 1482-1508.

First available in Project Euclid: 14 May 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35] 60H10: Stochastic ordinary differential equations [See also 34F05] 91G10: Portfolio theory

Brownian particle systems concentration of measure transportation cost inequalities Skorokhod maps stochastic portfolio theory Atlas model


Pal, Soumik; Shkolnikov, Mykhaylo. Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24 (2014), no. 4, 1482--1508. doi:10.1214/13-AAP954.

Export citation


  • [2] Arguin, L.-P. and Aizenman, M. (2009). On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 1080–1113.
  • [3] Banner, A. D., Fernholz, R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296–2330.
  • [4] Banner, A. D. and Ghomrasni, R. (2008). Local times of ranked continuous semimartingales. Stochastic Process. Appl. 118 1244–1253.
  • [5] Chatterjee, S. and Pal, S. (2010). A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 123–159.
  • [6] Chatterjee, S. and Pal, S. (2011). A combinatorial analysis of interacting diffusions. J. Theoret. Probab. 24 939–968.
  • [7] Dembo, A. (1997). Information inequalities and concentration of measure. Ann. Probab. 25 927–939.
  • [8] Dembo, A. and Zeitouni, O. (1996). Transportation approach to some concentration inequalities in product spaces. Electron. Commun. Probab. 1 83–90 (electronic).
  • [9] Djellout, H., Guillin, A. and Wu, L. (2004). Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 2702–2732.
  • [10] Dupuis, P. and Ishii, H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stoch. Stoch. Rep. 35 31–62.
  • [11] Dupuis, P. and Ramanan, K. (1999). Convex duality and the Skorokhod problem. I. Probab. Theory Related Fields 115 153–195.
  • [12] Dupuis, P. and Ramanan, K. (1999). Convex duality and the Skorokhod problem. II. Probab. Theory Related Fields 115 197–236.
  • [13] Fernholz, E. R. (2002). Stochastic Portfolio Theory. Springer, New York.
  • [14] Fernholz, R. and Karatzas, I. (2009). Stochastic portfolio theory: A survey. In Handbook of Numerical Analysis: Mathematical Modeling and Numerical Methods in Finance 89–168. Elsevier, Amsterdam.
  • [15] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.
  • [16] Ichiba, T. and Karatzas, I. (2010). On collisions of Brownian particles. Ann. Appl. Probab. 20 951–977.
  • [17] Ichiba, T., Papathanakos, V., Banner, A., Karatzas, I. and Fernholz, R. (2011). Hybrid atlas models. Ann. Appl. Probab. 21 609–644.
  • [18] Jourdain, B. and Malrieu, F. (2008). Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF. Ann. Appl. Probab. 18 1706–1736.
  • [19] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
  • [20] Marton, K. (1996). Bounding $\bar{d}$-distance by informational divergence: A method to prove measure concentration. Ann. Probab. 24 857–866.
  • [21] McKean, H. P. and Shepp, L. A. (2005). The advantage of capitalism vs. socialism depends on the criterion. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 328 160–168, 279–280.
  • [22] Pal, S. (2012). Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Related Fields 154 225–254.
  • [23] Pal, S. and Pitman, J. (2008). One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18 2179–2207.
  • [24] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • [25] Ruzmaikina, A. and Aizenman, M. (2005). Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 82–113.
  • [26] Shkolnikov, M. (2009). Competing particle systems evolving by i.i.d. increments. Electron. J. Probab. 14 728–751.
  • [27] Shkolnikov, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21 1911–1932.
  • [28] Shkolnikov, M. (2012). Large systems of diffusions interacting through their ranks. Stochastic Process. Appl. 122 1730–1747.
  • [29] Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1–34.
  • [30] Talagrand, M. (1996). Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 587–600.