×

One-dimensional Brownian particle systems with rank-dependent drifts. (English) Zbl 1166.60061

The paper studies interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate process at that time. Loose speaking, the Brownian particles evolve independently, except that the \(i\)’th-ranked particle is given drift \(\delta_i\). The main objective of the paper is to characterize the long-range behaviour of the spacings between the Brownian motions arranged in increasing order. For finite particle systems necessary and sufficient conditions on the drifts \(\delta_1,\dotsc,\delta_N\) for the spacing between the highest and the lowest particle to be tight over \(t \geq 0\) is given. In the affirmative case, the distribution of the spacings is shown to converge in total variation norm to a unique stationary distribution given by independent exponential variables with specified rates. In particular, this holds for the so-called Atlas model, which is given by \(\delta_1 = \delta > 0\) and \(\delta_2=\dotsc=\delta_N=0\). The second part of the paper studies the infinite dimensional Atlas model, i.e. countable infinite many Brownian motions interacting through ranks via \(\delta_1 = \delta > 0\) and \(0=\delta_2=\dotsc\). It is proven, that independent and identically distributed exponential spacings remain stationary under the dynamics of such a model. Some related conjectures in this direction are also discussed. The paper is thorough and detailed, e.g. discussing existence and uniqueness of such particle systems, and an extensive review of the literature is also given.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60G15 Gaussian processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Arratia, R. (1983). The motion of a tagged particle in the simple symmetric exclusion system on Z . Ann. Probab. 11 362-373. · Zbl 0515.60097
[2] Arratia, R. (1985). Symmetric exclusion processes: A comparison inequality and a large deviation result. Ann. Probab. 13 53-61. · Zbl 0558.60075
[3] Banner, A. D., Fernholz, R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296-2330. · Zbl 1099.91056
[4] Baryshnikov, Y. (2001). GUEs and queues. Probab. Theory Related Fields 119 256-274. · Zbl 0980.60042
[5] Chatterjee, S. and Pal, S. (2007). A phase transition behavior for Brownian motions interacting through their ranks. Submitted. · Zbl 1188.60049
[6] De Masi, A. and Ferrari, P. A. (2002). Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process. J. Statist. Phys. 107 677-683. · Zbl 1008.82022
[7] Dürr, D., Goldstein, S. and Lebowitz, J. L. (1985). Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Comm. Pure Appl. Math. 38 573-597. · Zbl 0578.60094
[8] Dürr, D., Goldstein, S. and Lebowitz, J. L. (1987). Self-diffusion in a nonuniform one-dimensional system of point particles with collisions. Probab. Theory Related Fields 75 279-290. · Zbl 0596.60056
[9] Durrett, R. (1996). Stochastic Calculus : A Practical Introduction . CRC Press, Boca Raton, FL. · Zbl 0856.60002
[10] Fernholz, E. R. (2002). Stochastic Portfolio Theory. Applications of Mathematics ( New York ) 48 . Springer, New York. · Zbl 1049.91067
[11] Fernholz, R. and Karatzas, I. (2007). Stochastic portfolio theory: A survey. Handb. Numer. Anal. · Zbl 1180.91267
[12] Ferrari, P. A. (1996). Limit theorems for tagged particles. Markov Process. Related Fields 2 17-40. Disordered systems and statistical physics: Rigorous results (Budapest, 1995). · Zbl 0879.60109
[13] Ferrari, P. A. and Fontes, L. R. G. (1994). The net output process of a system with infinitely many queues. Ann. Appl. Probab. 4 1129-1144. · Zbl 0812.60081
[14] Ferrari, P. A. and Fontes, L. R. G. (1996). Poissonian approximation for the tagged particle in asymmetric simple exclusion. J. Appl. Probab. 33 411-419. JSTOR: · Zbl 0855.60097
[15] Harris, T. E. (1965). Diffusion with “collisions” between particles. J. Appl. Probab. 2 323-338. JSTOR: · Zbl 0139.34804
[16] Harrison, J. M. (1973). The heavy traffic approximation for single server queues in series. J. Appl. Probab. 10 613-629. JSTOR: · Zbl 0287.60102
[17] Harrison, J. M. (2000). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab. 10 75-103. · Zbl 1131.60306
[18] Harrison, J. M. and Van Mieghem, J. A. (1997). Dynamic control of Brownian networks: State space collapse and equivalent workload formulations. Ann. Appl. Probab. 7 747-771. · Zbl 0885.60080
[19] Harrison, J. M. and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22 77-115. · Zbl 0632.60095
[20] Harrison, J. M. (1978). The diffusion approximation for tandem queues in heavy traffic. Adv. in Appl. Probab. 10 886-905. JSTOR: · Zbl 0387.60090
[21] Jourdain, B. and Malrieu, F. (2007). Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf. Ann. Appl. Probab. · Zbl 1185.65013
[22] Kipnis, C. (1986). Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14 397-408. · Zbl 0601.60098
[23] McKean, H. P. and Shepp, L. A. (2005). The advantage of capitalism vs. socialism depends on the criterion. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. ( POMI ) 328 160-168, 279-280. · Zbl 1121.91068
[24] O’Connell, N. and Yor, M. (2001). Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96 285-304. · Zbl 1058.60078
[25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0917.60006
[26] Rost, H. and Vares, M. E. (1985). Hydrodynamics of a one-dimensional nearest neighbor model. In Particle Systems , Random Media and Large Deviations ( Brunswick , Maine , 1984 ). Contemporary Mathematics 41 329-342. Amer. Math. Soc., Providence, RI. · Zbl 0572.60095
[27] Seppäläinen, T. (1997). A scaling limit for queues in series. Ann. Appl. Probab. 7 855-872. · Zbl 0897.60094
[28] Srinivasan, R. (1993). Queues in series via interacting particle systems. Math. Oper. Res. 18 39-50. JSTOR: · Zbl 0769.60085
[29] Sznitman, A.-S. (1986). A propagation of chaos result for Burgers’ equation. Probab. Theory Related Fields 71 581-613. · Zbl 0597.60055
[30] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX-1989. Lecture Notes in Mathematics 1464 165-251. Springer, Berlin. · Zbl 0732.60114
[31] Varadhan, S. R. S. and Williams, R. J. (1985). Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 405-443. · Zbl 0579.60082
[32] Williams, R. J. (1987). Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 459-485. · Zbl 0608.60074
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.