Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The infinite Atlas process: Convergence to equilibrium

Amir Dembo, Milton Jara, and Stefano Olla

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The semi-infinite Atlas process is a one-dimensional system of Brownian particles, where only the leftmost particle gets a unit drift to the right. Its particle spacing process has infinitely many stationary measures, with one distinguished translation invariant reversible measure. We show that the latter is attractive for a large class of initial configurations of slowly growing (or bounded) particle densities. Key to our proof is a new estimate on the rate of convergence to equilibrium for the particle spacing in a triangular array of finite, large size systems.

Résumé

Let modèle de Atlas demi-infini est un système unidimensional des particules Browniens, où seulement la particule plus à gauche a une vitesse positive. Le processus d’incréments correspondant a une infinité des mesures invariantes, avec une mesure distinguée, reversible et invariante par translations. On montre que cette mesure attire une grande classe des configurations initiales avec densité bornée ou à croissance modérée. Central à notre preuve est une nouvelle estimation de la vitesse de convergence vers l’équilibre du processus d’incréments dans un tableau triangulaire de systèmes finis et de grande taille.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 607-619.

Dates
Received: 11 September 2017
Revised: 15 November 2017
Accepted: 16 November 2017
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820825

Digital Object Identifier
doi:10.1214/17-AIHP875

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes [See also 58J65]

Keywords
Interacting particles Reflecting Brownian motions Non-equilibrium hydrodynamics Infinite Atlas process

Citation

Dembo, Amir; Jara, Milton; Olla, Stefano. The infinite Atlas process: Convergence to equilibrium. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 607--619. doi:10.1214/17-AIHP875. https://projecteuclid.org/euclid.aihp/1557820825


Export citation

References

  • [1] L. Ambrosio and G. Savaré. Gradient flows of probability measures. In Handbook of Evolution Equations (III). Elsevier, Amsterdam, 2006.
  • [2] R. Arratia. The motion of tagged particle in the simple exclusion system in $\mathbb{Z}$. Ann. Probab. 11 (1983) 362–373.
  • [3] R. Bass and E. Pardoux. Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 (1987) 557–572.
  • [4] M. Cabezas, A. Dembo, A. Sarantsev and V. Sidoravicius. Brownian particles with rank-dependent drifts: Out-of-equilibrium behavior, 2017. Available at arXiv:1708.01918.
  • [5] A. Dembo and L. Tsai. Equilibrium fluctuation of the Atlas model. Ann. Probab. 45 (6B) (2017) 4529–4560.
  • [6] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. I. Comm. Pure Appl. Math. 28 (1975) 1–47.
  • [7] P. Dupuis and R. J. Williams. Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 (1994) 680–702.
  • [8] D. Dürr, S. Goldstein and J. L. Lebowitz. Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Comm. Pure Appl. Math. 38 (1985) 573–597.
  • [9] E. R. Fernholz and I. Karatzas. Stochastic portfolio theory: An overview. In Handbook of Numerical Analysis 15 89–167. North-Holland, Amsterdam, 2009.
  • [10] T. E. Harris. Diffusion with “collisions” between particles. J. Appl. Probab. 2 (1965) 323–338.
  • [11] J. M. Harrison and R. J. Williams. Multidimensional reflected Brownian motions having exponential stationary sistributions. Ann. Probab. 15 (1987) 115–137.
  • [12] F. Hernández, M. Jara and F. J. Valentim. Equilibrium fluctuations for a discrete Atlas model. Stochastic Process. Appl. 127 (2017) 783–802.
  • [13] T. Ichiba, I. Karatzas and M. Shkolnikov. Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156 (2013) 229–248.
  • [14] T. Ichiba, S. Pal and M. Shkolnikov. Convergence rates for rank-based models with applications to portfolio theory. Probab. Theory Related Fields 156 (2013) 415–448.
  • [15] S. Pal and J. Pitman. One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18 (2008) 2179–2207.
  • [16] H. Rost and M. E. Vares. Hydrodynamics of a one dimensional nearest neighbor model. Contemp. Math. 41 (1985) 329–342.
  • [17] A. Ruzmaikina and M. Aizenman. Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 (2005) 82–113.
  • [18] A. Sarantsev. Comparison techniques for competing Brownian particles. J. Theoret. Probab. To appear. Available at arXiv:1305.1653.
  • [19] A. Sarantsev. Two-sided infinite systems of competing Brownian particles. ESAIM Probab. Stat. 21 (2015) 317–349.
  • [20] A. Sarantsev. Infinite systems of competing Brownian particles. Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 2279–2315.
  • [21] A. Sarantsev and L. Tsai. Stationary gap distributions for infinite systems of competing Brownian particles. Electron. J. Probab. 22 (56) (2017) 1–20.
  • [22] M. Shkolnikov. Competing particle systems evolving by I.I.D. increments. Electron. J. Probab. 14 (27) (2009) 728–751.
  • [23] M. Shkolnikov. Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21 (2011) 1911–1932.
  • [24] L. Tsai. Stationary distributions of the Atlas model. Electron. Commun. Probab. 23 (10) (2018) 1–10.
  • [25] R. J. Williams. Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 (1987) 459–485.
  • [26] R. J. Williams. Semimartingale reflecting Brownian motions in the orthant. Stochastic networks. IMA Vol. Math. Appl. 71 (1995) 125–137.