Electronic Journal of Probability

Brownian motions with one-sided collisions: the stationary case

Patrik Ferrari, Herbert Spohn, and Thomas Weiss

Full-text: Open access


We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the limit $t\to\infty$. This leads to a new universal cross-over process.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 69, 41 pp.

Accepted: 23 June 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]

Reflected Brownian motion KPZ universality class

This work is licensed under aCreative Commons Attribution 3.0 License.


Ferrari, Patrik; Spohn, Herbert; Weiss, Thomas. Brownian motions with one-sided collisions: the stationary case. Electron. J. Probab. 20 (2015), paper no. 69, 41 pp. doi:10.1214/EJP.v20-4177. https://projecteuclid.org/euclid.ejp/1465067175

Export citation


  • Amir, Gideon; Corwin, Ivan; Quastel, Jeremy. Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math. 64 (2011), no. 4, 466–537.
  • Anderson, Robert F.; Orey, Steven. Small random perturbation of dynamical systems with reflecting boundary. Nagoya Math. J. 60 (1976), 189–216.
  • Baik, Jinho; Buckingham, Robert; DiFranco, Jeffery. Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function. Comm. Math. Phys. 280 (2008), no. 2, 463–497.
  • Baik, Jinho; Ferrari, Patrik L.; Péché, Sandrine. Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math. 63 (2010), no. 8, 1017–1070.
  • Baik, Jinho; Rains, Eric M. Limiting distributions for a polynuclear growth model with external sources. J. Statist. Phys. 100 (2000), no. 3-4, 523–541.
  • Baik, Jinho; Wang, Dong. On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank 1 case. Int. Math. Res. Not. IN 2011, no. 22, 5164–5240.
  • A. Borodin, Private communication, (2008).
  • Borodin, Alexei; Corwin, Ivan; Ferrari, Patrik. Free energy fluctuations for directed polymers in random media in $1+1$ dimension. Comm. Pure Appl. Math. 67 (2014), no. 7, 1129–1214.
  • Borodin, Alexei; Corwin, Ivan; Remenik, Daniel. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 1, 28–58.
  • Borodin, Alexei; Ferrari, Patrik L. Large time asymptotics of growth models on space-like paths. I. PushASEP. Electron. J. Probab. 13 (2008), no. 50, 1380–1418.
  • Borodin, Alexei; Ferrari, Patrik L.; Prähofer, Michael; Sasamoto, Tomohiro. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007), no. 5-6, 1055–1080.
  • Corwin, Ivan; Quastel, Jeremy; Remenik, Daniel. Continuum statistics of the ${\rm Airy}_ 2$ process. Comm. Math. Phys. 317 (2013), no. 2, 347–362.
  • Ferrari, Patrik L.; Frings, René. Perturbed GUE minor process and Warren's process with drifts. J. Stat. Phys. 154 (2014), no. 1-2, 356–377.
  • Ferrari, Patrik L.; Spohn, Herbert. Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 (2006), no. 1, 1–44.
  • P.L. Ferrari, H. Spohn, and T. Weiss, Scaling Limit for Brownian Motions with One-sided Collisions, arXiv:1306.5095 (2013).
  • T. Funaki and J. Quastel, KPZ equation, its renormalization and invariant measures, arXiv:1407.7310 (2014).
  • Gravner, Janko; Tracy, Craig A.; Widom, Harold. Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 (2001), no. 5-6, 1085–1132.
  • Hairer, Martin. Solving the KPZ equation. Ann. of Math. (2) 178 (2013), no. 2, 559–664.
  • Imamura, T.; Sasamoto, T. Fluctuations of the one-dimensional polynuclear growth model with external sources. Nuclear Phys. B 699 (2004), no. 3, 503–544.
  • Imamura, Takashi; Sasamoto, Tomohiro. Stationary correlations for the 1D KPZ equation. J. Stat. Phys. 150 (2013), no. 5, 908–939.
  • Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
  • Johansson, Kurt. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003), no. 1-2, 277–329.
  • Johansson, Kurt. Two time distribution in Brownian directed percolation, arXiv:1502.00941 (2015).
  • M. Kardar, G. Parisi, and Y.Z. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889–892.
  • Ledoux, M. Deviation inequalities on largest eigenvalues. Geometric aspects of functional analysis, 167–219, Lecture Notes in Math., 1910, Springer, Berlin, 2007.
  • O'Connell, Neil; Yor, Marc. Brownian analogues of Burke's theorem. Stochastic Process. Appl. 96 (2001), no. 2, 285–304.
  • Prähofer, Michael; Spohn, Herbert. Scale invariance of the PNG droplet and the Airy process. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Statist. Phys. 108 (2002), no. 5-6, 1071–1106.
  • Prähofer, Michael; Spohn, Herbert. Exact scaling functions for one-dimensional stationary KPZ growth. J. Statist. Phys. 115 (2004), no. 1-2, 255–279.
  • T. Seppäläinen and B. Valkó, Bounds for scaling exponents for a 1+1 dimensional directed polymer in a brownian environment, ALEA, to appear; arXiv:1006.4864.
  • A.V. Skorokhod, Stochastic equations for diffusions in a bounded region, Theory Probab. Appl. (1961), 264–274.
  • Spohn, Herbert. Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154 (2014), no. 5, 1191–1227.
  • G. Szegö, Orthogonal polynomials, 3th ed., American Mathematical Society Providence, Rhode Island, 1967.
  • Warren, Jon. Dyson's Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 (2007), no. 19, 573–590.
  • T. Weiss, Scaling behaviour of the directed polymer model of Baryshnikov and O'Connell at zero temperature, Bachelor Thesis, TU-München (2011).