×

Modeling flocks and prices: jumping particles with an attractive interaction. (English. French summary) Zbl 1302.60130

Consider \(n\) particles on the real line. The position in \({\mathbb R}\) of the particles at time \(t\) is denoted by \(x_1(t )\), \(x_2(t ), \dots{} , x_n(t)\), respectively. The mean position of the particles is \(m_n(t) = \frac 1 n \sum_{i=1}^n x_i (t)\). Next, let \(w:{\mathbb R}\to {\mathbb R}_+\) be a positive and monotone decreasing function. We now describe the dynamics which is a continuous-time Markov jump process. The \(i\)th particle, positioned at \(x_i (t)\) at time \(t\), jumps with rate \(w(x_i (t) - m_n(t))\). That is, conditioned on the configuration of the particles, jumps happen independently after an exponentially distributed time, with the parameter of the \(i\)th particle being \(w(x_i (t) - m_n(t))\). When a particle jumps, the length of the jump is a random positive number \(Z\) from a specific distribution. This is independent of time and position of the particle and of all other particles as well. Assume that \({\mathbb E}Z = 1\), and \(Z\) has a finite third moment. The authors focus on the empirical measure of the \(n\) particle process at time \(t\): \(\mu_n(t) =\frac 1 n \sum_{i=1}^n \delta_{x_i (t)}\). The main result of this paper says that, under suitable conditions, as \(n\to\infty\), \(\mu_n\) converges in the Skorokhod space \(D([0, \infty), {\mathcal P}_1(\mathbb R))\) to some \(\mu\), where \({\mathcal P}_1(\mathbb R)\) is the space of probabilities having finite first moment, and \(\mu\) is the unique solution to the mean field equation \[ \langle f, \mu (t)\rangle - \langle f, \mu (0)\rangle -\int_0^t \Big[{\mathbb E}\big(f(x+Z)-f(x)\big) w\big(x-\mu(s)(\mathbb R)\big)\Big] \mu(s) \,ds \] for every \(t\geq 0\) and a class of test functions \(f\). Here, \(\langle f, \mu \rangle=\int f \,d\mu\), as usual.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J75 Jump processes (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv arXiv Euclid

References:

[1] L. P. Arguin. Competing particle systems and the Ghirlanda-Guerra identities. Electron. J. Probab. 13 (2008) 2101-2117. · Zbl 1192.60103 · doi:10.1214/EJP.v13-579
[2] L. P. Arguin and M. Aizenman. On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 (2009) 1080-1113. · Zbl 1177.60050 · doi:10.1214/08-AOP429
[3] M. Balázs, M. Z. Rácz, and B. Tóth. Modeling flocks and prices: Jumping particles with an attractive interaction. Preprint, 2011. Available at . 1107.3289
[4] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl. Acad. Sci. USA 105 (2008) 1232-1237.
[5] A. D. Banner, R. Fernholz and I. Karatzas. Atlas models of equity markets. Ann. Appl. Probab. 15 (2005) 2296-2330. · Zbl 1099.91056 · doi:10.1214/105051605000000449
[6] D. Ben-Avraham, S. N. Majumdar and S. Redner. A toy model of the rat race. J. Stat. Mech. Theory Exp. 2007 (2007) L04002.
[7] E. Bertin. Global fluctuations and Gumbel statistics. Phys. Rev. Lett. 95 (2005) 170601.
[8] P. Billingsley. Convergence of Probability Measures , 2nd edition. Wiley Series in Probability and Statistics . Wiley, New York, 1999. · Zbl 0944.60003
[9] S. Chatterjee and S. Pal. A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 (2009) 123-159. · Zbl 1188.60049 · doi:10.1007/s00440-009-0203-0
[10] M. Clusel and E. Bertin. Global fluctuations in physical systems: A subtle interplay between sum and extreme value statistics. Internat. J. Modern Phys. B 22 (2008) 3311-3368. · Zbl 1145.82304 · doi:10.1142/S021797920804853X
[11] A. Czirók, A. L. Barabási and T. Vicsek. Collective motion of self-propelled particles: Kinetic phase transition in one dimension. Phys. Rev. Lett. 82 (1999) 209-212.
[12] J. Engländer. The center of mass for spatial branching processes and an application for self-interaction. Electron. J. Probab. 15 (2010) 1938-1970. · Zbl 1226.60118
[13] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence . Wiley, New York, 1986. · Zbl 0592.60049
[14] J. Feng and T. G. Kurtz. Large Deviations for Stochastic Processes , Mathematical Surveys and Monographs 131 . Amer. Math. Soc., Providence, RI, 2006.
[15] E. R. Fernholz. Stochastic Portfolio Theory . Springer, New York, 2002. · Zbl 1049.91067
[16] E. R. Fernholz and I. Karatzas. Stochastic portfolio theory: An overview. In Handbook of Numerical Analysis 15 89-167. Elsevier, Amsterdam, 2009. · Zbl 1180.91267 · doi:10.1016/S1570-8659(08)00003-3
[17] A. L. Gibbs and F. E. Su. On choosing and bounding probability metrics. International Statistical Review 70 (2002) 419-435. · Zbl 1217.62014 · doi:10.2307/1403865
[18] A. G. Greenberg, V. A. Malyshev and S. Y. Popov. Stochastic models of massively parallel computation. Markov Process. Related Fields 1 (1995) 473-490. · Zbl 0902.60072
[19] A. Greven and F. D. Hollander. Phase transitions for the long-time behaviour of interacting diffusions. Ann. Probab. 35 (2007) 1250-1306. · Zbl 1126.60085 · doi:10.1214/009117906000001060
[20] I. Grigorescu and M. Kang. Steady state and scaling limit for a traffic congestion model. ESAIM Probab. Stat. 14 (2010) 271-285. · Zbl 1227.60108 · doi:10.1051/ps:2008029
[21] E. J. Gumbel. Statistics of Extremes . Dover, New York, 1958. · Zbl 0086.34401
[22] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes , 2nd edition. Springer, Berlin, 2003. · Zbl 1018.60002
[23] P. M. Kotelenez and T. G. Kurtz. Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type. Probab. Theory Related Fields 146 (2010) 189-222. · Zbl 1189.60123 · doi:10.1007/s00440-008-0188-0
[24] A. Manita and V. Shcherbakov. Asymptotic analysis of a particle system with mean-field interaction. Markov Process. Related Fields 11 (2005) 489-518. · Zbl 1099.60073
[25] S. Pal and J. Pitman. One-dimensional Brownian particle systems with rank dependent drifts. Ann. Appl. Probab. 18 (2008) 2179-2207. · Zbl 1166.60061 · doi:10.1214/08-AAP516
[26] E. A. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999) 125-324 Lecture. Notes in Math. 1781 . Springer, Berlin, 2002. · Zbl 1020.60075
[27] A. Ruzmaikina and M. Aizenman. Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 (2005) 82-113. · Zbl 1096.60042 · doi:10.1214/009117904000000865
[28] M. Shkolnikov. Competing particle systems evolving by IID increments. Electron. J. Probab. 14 (2009) 728-751. · Zbl 1190.60039 · doi:10.1214/EJP.v14-635
[29] M. Shkolnikov. Competing particle systems evolving by interacting Levy processes. Ann. Appl. Probab. 21 (2011) 1911-1932. · Zbl 1238.60113 · doi:10.1214/10-AAP743
[30] M. Shkolnikov. Large volatility-stabilized markets. Preprint, 2011. Available at . 1102.3461 · Zbl 1288.60092 · doi:10.1016/j.spa.2012.09.001
[31] M. Shkolnikov. Large systems of diffusions interacting through their ranks. Stochastic Process. Appl. 122 (2012) 1730-1747. · Zbl 1276.60087 · doi:10.1016/j.spa.2012.01.011
[32] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75 (1995) 1226-1229. · Zbl 0869.92002
[33] S. Willard. General Topology . Dover, New York, 2004. · Zbl 1052.54001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.