Electronic Journal of Probability

Stationary gap distributions for infinite systems of competing Brownian particles

Andrey Sarantsev and Li-Cheng Tsai

Full-text: Open access


Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) [PP08] in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.

Article information

Electron. J. Probab. Volume 22 (2017), paper no. 56, 20 pp.

Received: 1 February 2017
Accepted: 18 June 2017
First available in Project Euclid: 5 July 2017

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

competing Brownian particles infinite Atlas model stationary distribution gap process

Creative Commons Attribution 4.0 International License.


Sarantsev, Andrey; Tsai, Li-Cheng. Stationary gap distributions for infinite systems of competing Brownian particles. Electron. J. Probab. 22 (2017), paper no. 56, 20 pp. doi:10.1214/17-EJP78. https://projecteuclid.org/euclid.ejp/1499220068

Export citation


  • [Ald03] David Aldous (2003). Unpublished, available athttp://www.stat.berkeley.edu/~aldous/Research/OP/river.pdf
  • [AA09] Louis-Pierre Arguin, Michael Aizenman (2009). On the Structure of Quasi-Stationary Competing Particle Systems.Ann. Probab.37(3), 1080–1113.
  • [BFK05] Adrian D. Banner, E. Robert Fernholz, Ioannis Karatzas (2005). Atlas Models of Equity Markets.Ann. Appl. Probab.15(4), 2996–2330.
  • [BG08] Adrian D. Banner, Raouf Ghomrasni (2008). Local Times of Ranked Continuous Semimartingales.Stoch. Proc. Appl.118(7), 1244–1253.
  • [BFIKP11] Adrian D. Banner, E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas, Vassilios Papathanakos (2011). Hybrid Atlas Models.Ann. Appl. Probab.21(2), 609–644.
  • [BP87] Richard Bass, E. Pardoux (1987). Uniqueness for Diffusions with Piecewise Constant Coefficients.Probab. Th. Rel. Fields76, 557–572.
  • [BS15] Cameron Bruggeman, Andrey Sarantsev (2015). Multiple Collisions in Systems of Competing Brownian Particles. To appear inBernoulli. Available atarXiv:1309.2621.
  • [CP10] Sourav Chatterjee, Soumik Pal (2010). A Phase-Transition Behavior for Brownian Motions Interacting Through Their Ranks.Probab. Th. Rel. Fields147(1–2), 123–159.
  • [DSVZ16] Amir Dembo, Mykhaylo Shkolnikov, S.R. Srinivasa Varadhan, Ofer Zeitouni (2016). Large Deviations for Diffusions Interacting Through Their Ranks.Comm. Pure Appl. Math.69(7), 1259–1313.
  • [DT15] Amir Dembo, Li-Cheng Tsai (2015). Equilibrium Fluctuations of the Atlas Model. To appear inAnn. Probab.Available atarXiv:1503.03581.
  • [Fel68] William Feller (1968).An Introduction to Probability Theory and Its Applications.Vol. 1, Wiley.
  • [FIK13] E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas (2013). A Second-Order Stock Market Model.Ann. Finance9(3), 439–454.
  • [FIKP13] E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas, Vilmos Prokaj (2013). Planar Diffusions with Rank-Based Characteristics and Perturbed Tanaka Equations.Probab. Th. Rel. Fields,156(1–2), 343–374.
  • [FIK13] E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas (2013). Two Brownian Particles with Rank-Based Characteristics and Skew-Elastic Collisions.Stoch. Proc. Appl.123(8), 2999–3026.
  • [FK09] E. Robert Fernholz, Ioannis Karatzas (2009) Stochastic Portfolio Theory: An Overview.Handbook of Numerical Analysis: Mathematical Modeling and Numerical Methods in Finance, 89–168. Elsevier.
  • [IK10] Tomoyuki Ichiba, Ioannis Karatzas (2010). On Collisions of Brownian Particles.Ann. Appl. Probab.20(3), 951–977.
  • [IKP13] Tomoyuki Ichiba, Ioannis Karatzas, Vilmos Prokaj (2013). Diffusions with Rank-Based Characteristics and Values in the Nonnegative Quadrant.Bernoulli19(5B), 2455–2493.
  • [IKS13] Tomoyuki Ichiba, Ioannis Karatzas, Mykhaylo Shkolnikov (2013). Strong Solutions of Stochastic Equations with Rank-Based Coefficients.Probab. Th. Rel. Fields156, 229–248.
  • [IPS13] Tomoyuki Ichiba, Soumik Pal, Mykhaylo Shkolnikov (2013). Convergence Rates for Rank-Based Models with Applications to Portfolio Theory.Probab. Th. Rel. Fields156, 415–448.
  • [JM08] Benjamin Jourdain, Florent Malrieu (2008). Propagation of Chaos and Poincare Inequalities for a System of Particles Interacting Through Their cdf.Ann. Appl. Probab.18(5), 1706–1736.
  • [JR13] Benjamin Jourdain, Julien Reygner (2013). Propagation of Chaos for Rank-Based Interacting Diffusions and Long Time Behaviour of a Scalar Quasilinear Parabolic Equation.SPDE Anal. Comp.1(3), 455–506.
  • [JR14] Benjamin Jourdain, Julien Reygner (2014). The Small Noise Limit of Order-Based Diffusion Processes.Electr. J. Probab.19(29), 1–36.
  • [JR15] Benjamin Jourdain, Julien Reygner (2015). Capital Distribution and Portfolio Performance in the Mean-Field Atlas Model.Ann. Finance11(2), 151–198.
  • [KPS12] Ioannis Karatzas, Soumik Pal, Mykhaylo Shkolnikov (2016). Systems of Brownian Particles with Asymmetric Collisions.Ann. Inst. H. Poincare52(1), 323–354.
  • [KS16] Ioannis Karatzas, Andrey Sarantsev (2016). Diverse Market Models of Competing Brownian Particles with Splits and Mergers.Ann. Appl. Probab.26(3), 1329–1361.
  • [PP08] Soumik Pal, Jim Pitman (2008). One-Dimensional Brownian Particle Systems with Rank-Dependent Drifts.Ann. Appl. Probab.18(6), 2179–2207.
  • [Rey15] Julien Reygner (2015). Chaoticity of the Stationary Distribution of Rank-Based Interacting Diffusions.Electr. Comm. Probab.20(60), 1–20.
  • [RA05] Anastasia Ruzmaikina, Michael Aizenman (2005). Characterization of Invariant Measures at the Leading Edge for Competing Particle Systems.Ann. Probab.33(1), 82–113.
  • [Sar15] Andrey Sarantsev (2015). Comparison Techniques for Competing Brownian Particles. To appear inJ. Th. Probab.Available atarXiv:1305.1653.
  • [Sar15a] Andrey Sarantsev (2015). Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution. To appear inJ. Th. Probab.Available atarXiv:1509.01781.
  • [Sar15b] Andrey Sarantsev (2015). Triple and Simultaneous Collisions of Competing Brownian Particles.Electr. J. Probab.20(29), 1–28.
  • [Sar15c] Andrey Sarantsev (2015). Two-Sided Infinite Systems of Competing Brownian Particles. To appear inESAIM P&S.Available atarXiv:1509.01859.
  • [Sar16] Andrey Sarantsev (2016). Explicit Rates of Exponential Convergence for Reflected Jump-Diffusions on the Half-Line.ALEA Lat. Am. J. Probab. Math. Stat.16(2), 1069–1093.
  • [Sar16a] Andrey Sarantsev (2016). Infinite Systems of Competing Brownian Particles. To appear inAnn. Inst. H. Poincare.Available atarXiv:1403.4229.
  • [Shk11] Mykhaylo Shkolnikov (2011). Competing Particle Systems Evolving by Interacting Lévy Processes.Ann. Appl. Probab.21(5), 1911–1932.
  • [Shk12] Mykhaylo Shkolnikov (2012). Large Systems of Diffusions Interacting Through Their Ranks.Stoch. Proc. Appl.122(4), 1730–1747.
  • [Str11] Daniel W. Stroock (2011).Probability Theory. An Analytic View. Cambridge University Press.
  • [Tsa17] Li-Cheng Tsai (2017). Stationary Distributions of the Atlas Model. Available atarXiv:1702.02043.
  • [Wil95] Ruth J. Williams (1995).Semimartingale Reflecting Brownian Motions in the Orthant. Stochastic networks, IMA Vol. Math. Appl.71, 125–137. Springer-Verlag.