Electronic Journal of Probability

Invariant measures of mass migration processes

Abstract

We introduce the “mass migration process” (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\geq 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We derive necessary and sufficient conditions for the existence of translation invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation invariant, invariant probability measures. We also consider attractiveness properties (through couplings), condensation phenomena, and their links for MMP. We illustrate our results on many examples; we prove the coexistence of condensation and attractiveness in one of them.

Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 60, 52 pp.

Dates
Accepted: 20 July 2016
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.ejp/1475266508

Digital Object Identifier
doi:10.1214/16-EJP4399

Citation

Fajfrová, Lucie; Gobron, Thierry; Saada, Ellen. Invariant measures of mass migration processes. Electron. J. Probab. 21 (2016), paper no. 60, 52 pp. doi:10.1214/16-EJP4399. https://projecteuclid.org/euclid.ejp/1475266508.

References

• [1] Andjel, E.D.: Invariant Measures for the Zero Range Process. Ann. Probab. 10 (1982), no. 3, 525–547.
• [2] Armendáriz, I., Loulakis, M.: Thermodynamic Limit for the Invariant Measures in Supercritical Zero Range Processes. Probab. Theory Related Fields, 145 (2009) no. 1–2, 175–188.
• [3] Barraquand, G., Corwin, I.: The $q$-Hahn asymmetric exclusion process. To appear in Ann. Appl. Probab. arXiv:1501.03445
• [4] Beltran, J., Landim, C.: Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Related Fields, 152 (2012), no. 3–4, 781–807.
• [5] Borrello, D.: Stochastic order and attractiveness for particle systems with multiple births, deaths and jumps. Electron. J. Probab. 16 (2011), no. 4, 106–151.
• [6] Borrello, D.: On the role of Allee effect and mass migration in survival and extinction of a species. Ann. Appl. Probab. 22 (2012), no. 2, 670–701.
• [7] Chleboun P., Großkinsky S.: Condensation in stochastic particle systems with stationary product measures. J. Stat. Phys. 154 (2014), no. 1–2, 432–465.
• [8] Cocozza, C. T.: Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 4, 509–523.
• [9] Evans, M. R.: Phase transitions in one-dimensional nonequilibrium systems. Braz. J. Phys. 30, no. 1 (2000), 42–57.
• [10] Evans, M. R. and Hanney, T.: Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models. J. Phys. A: Math. Gen. 38 (2005), no. 19, R195–R240.
• [11] Evans, M. R., Majumdar, S.N. and Zia, R.K.P.: Factorized steady states in mass transport models. J. Phys. A: Math. Gen. 37 (2004), no. 25, L275–L280.
• [12] Evans, M. R., Majumdar, S.N. and Zia, R.K.P.: Factorised steady states in mass transport models on an arbitrary graph. J. Phys. A 39 (2006), no. 18, 4859–4873.
• [13] Ferrari, P.A., Landim, C., Sisko, V.: Condensation for a fixed number of independent random variables. J. Stat. Phys. 128 (2007), no. 5, 1153–1158.
• [14] Gobron, T., Saada, E.: Couplings, attractiveness and hydrodynamics for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 46, no. 4 (2010), 1132–1177.
• [15] Godrèche, C.: From urn models to zero-range processes: statics and dynamics. Ageing and the glass transition, 261–294, Lect. Notes Phys. 716, Springer, Berlin, 2007.
• [16] Greenblatt, R.L., Lebowitz, J.L.: Product Measure Steady States of Generalized Zero Range Processes. J. Phys. A: Math. Gen. 39 (2006), no. 7, 1565–1573.
• [17] Großkinsky S., Schütz G.M. and Spohn H.: Condensation in the zero range process: stationary and dynamical properties. J. Statist. Phys., 113 (2003), no. 3–4, 389–410.
• [18] Liggett, T. M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56 (1981), no. 4, 443 – 468.
• [19] Liggett, T.M.: Interacting Particle Systems. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005.
• [20] Luck, J.M., Godrèche, C.: Nonequilibrium stationary states with Gibbs measure for two and three species of interacting particles. J. Stat. Mech. Theory Exp. (2006), no. 8, P08009, 18 pp.
• [21] Luck, J.M., Godrèche, C.: Structure of the stationary state of the asymmetric target process J. Stat. Mech. Theory Exp. (2007), no. 8, P08005, 30 pp.
• [22] Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales. Vol. 1. Foundations. Reprint of the second (1994) edition. Cambridge University Press, UK, 2000.
• [23] Rafferty T., Chleboun P., Großkinsky S.: Monotonicity and condensation in homogeneous stochastic particle systems. Preprint (2015). arXiv:1505.02049
• [24] Schütz, G. M., Ramaswamy, R., Barma, M.: Pairwise balance and invariant measures for generalized exclusion processes. J. Phys. A: Math. Gen. 29 (1996), no. 4, 837 – 843.
• [25] Seppäläinen, T.: A microscopic model for the Burgers equation and longest increasing subsequences. Electron. J. Probab. 1 (1996), no. 5, approx. 51 pp.
• [26] Spitzer, F.: Interaction of Markov Processes. Advances in Math. 5 (1970), 247–290.