Two-Sided Infinite Systems of Competing Brownian Particles

Sarantsev, Andrey

doi:10.1051/ps/2017013

Sarantsev, Andrey ¹

¹ Department of Statistics and Applied Probability, University of California, Santa BarbaraUSA

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 317-349.

Résumé

Two-sided infinite systems of Brownian particles with rank-dependent dynamics, indexed by all integers, exhibit different properties from their one-sided infinite counterparts, indexed by positive integers, and from finite systems. Consider the gap process, which is formed by spacings between adjacent particles. In stark contrast with finite and one-sided infinite systems, two-sided infinite systems can have one- or two-parameter family of stationary gap distributions, or the gap process weakly converging to zero as time goes to infinity.

Reçu le : 2017-02-10
Accepté le : 2017-07-20

MR Zbl

DOI : 10.1051/ps/2017013

Classification : 60J60, 60J55, 60J65, 60H10, 60K35
Mots-clés : Competing Brownian particles, gap process, weak convergence, stationary distribution, named particles, ranked particles, stochastic domination, interacting particle systems

Affiliations des auteurs :

Sarantsev, Andrey ¹

¹ Department of Statistics and Applied Probability, University of California, Santa BarbaraUSA

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Sarantsev, Andrey. Two-Sided Infinite Systems of Competing Brownian Particles. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 317-349. doi : 10.1051/ps/2017013. http://archive.numdam.org/articles/10.1051/ps/2017013/

Bibliographie
Cité par

L.-P. Arguin and M. Aizenman, On the Structure of Quasi-Stationary Competing Particle Systems. Ann. Probab. 37 (2009) 1080–1113. | MR | Zbl

R. Arratia, The Motion of a Tagged Particle in the Simple Symmetric Exclusion system on $Z$ . Ann. Probab. 11 (1983) 362–373. | DOI | MR | Zbl

A.D. Banner, E.R. Fernholz and I. Karatzas, Atlas Models of Equity Markets. Ann. Appl. Probab. 15 (2005) 2996–2330. | DOI | MR | Zbl

A.D. Banner, E.R. Fernholz, T. Ichiba and I. Karatzas, Vassilios Papathanakos. Hybrid Atlas Models. Ann. Appl. Probab. 21 (2011) 609–644. | MR | Zbl

Y. Baryshnikov, GUEs and Queues. Probab. Theory Relat. Fields 119 (2001) 256–274. | DOI | MR | Zbl

R. Bass and E. Pardoux, Uniqueness for Diffusions with Piecewise Constant Coefficients. Probab. Theory Relat. Fields 76 (1987) 557–572. | DOI | MR | Zbl

Maury Bramson, Thomas M. Liggett. Exclusion Processes in Higher Dimensions: Stationary Measures and Convergence. Ann. Probab. 33 (2005) 2255–2313. | MR | Zbl

M. Bramson and Th.M. Liggett, Characterization of Stationary Measures for One-Dimensional Exclusion Processes. Ann. Probab. 30 (2002) 1539–1575. | MR | Zbl

C. Bruggeman and A. Sarantsev, Multiple Collisions in Systems of Competing Brownian Particles. Bernoulli 24 (2018) 156–201. | DOI | MR | Zbl

S. Chatterjee and S. Pal, A Phase Transition Behavior for Brownian Motions Interacting Through Their Ranks. Probab. Theory Relat. Fields 147 (2010) 123–159. | DOI | MR | Zbl

J.G. Dai and R.J. Williams, Existence and Uniqueness of Semimartingale Reflecting Brownian Motions in Convex Polyhedrons. Theory Probab. Appl. 40 (1995) 1–40. | DOI | MR | Zbl

A. Dembo, M. Shkolnikov and S.R.S. Varahna and O. Zeitouni, Large Deviations for Diffusions Interacting Through Their Ranks. Comm. Pure Appl. Math. 69 (2016) 1259–1313. | DOI | MR | Zbl

A. Dembo and L.-Ch. Tsai, Equilibrium Fluctuations of the Atlas Model. Preprint (2015). To appear in Ann. Probab. (2018). | arXiv | MR

E.R. Fernholz, Stochastic Portfolio Theory. Appl. Math. 48. Springer (2002). | MR | Zbl

E.R. Fernholz, T. Ichiba and I. Karatzas, A Second-Order Stock Market Model. Ann. Fin. 9 (2013) 439–454. | DOI | MR | Zbl

E.R. Fernholz and I. Karatzas, Stochastic Portfolio Theory: an Overview. Handbook of Numerical Analysis: Mathematical Modeling and Numerical Methods Finance. Elsevier (2009) 89–168. | Zbl

E.R. Fernholz, T. Ichiba, I. Karatzas and V. Prokaj, Planar Diffusions with Rank-Based Characteristics and Perturbed Tanaka Equations. Probab. Theory Relat. Fields 156 (2013) 343–374. | DOI | MR | Zbl

P.A. Ferrari, Limit Theorems for Tagged Particles. Markov Proc. Relat. Fields 2 (1996) 17–40. | MR | Zbl

P.A. Ferrari, L. Renato and G. Fontes, The Net Output Process of a System with Infinitely Many Queues. Ann. Appl. Probab. 4 (1994) 1129–1144. | DOI | MR | Zbl

P.L. Ferrari, H. Spohn and Th. Weiss, Scaling Limit for Brownian Motions with One-Sided Collisions. Ann. Appl. Probab. 25 (2015) 1349–1382. | DOI | MR | Zbl

Th.E. Harris, Diffusions with Collisions Between Particles. J. Appl. Probab. 2 (1965) 323–338. | DOI | MR | Zbl

T. Ichiba and I. Karatzas, On Collisions of Brownian Particles. Ann. Appl. Probab. 20 (2010) 951–977. | DOI | MR | Zbl

T. Ichiba, I. Karatzas and M. Shkolnikov, Strong Solutions of Stochastic Equations with Rank-Based Coefficients. Probab. Theory Relat. Fields 156 (2013) 229–248. | DOI | MR | Zbl

T. Ichiba, S. Pal and M. Shkolnikov, Convergence Rates for Rank-Based Models with Applications to Portfolio Theory. Probab. Theory Relat. Fields 156 (2013) 415–448. | DOI | MR | Zbl

T. Ichiba and A. Sarantsev, Yet Another Condition for Absence of Collisions for Competing Brownian Particles. Electr. Commun. Probab. 22 (2017) 1–7. | MR | Zbl

B. Jourdain and F. Malrieu, Propagation of Chaos and Poincare Inequalities for a System of Particles Interacting Through Their Cdf. Ann. Appl. Probab. 18 (2008) 1706–1736. | DOI | MR | Zbl

B. Jourdain and J. Reygner, Propagation of Chaos for Rank-Based Interacting Diffusions and Long Time Behaviour of a Scalar Quasilinear Parabolic Equation. SPDE Anal. Comput. 1 (2013) 455–506. | MR | Zbl

B. Jourdain and J. Reygner, The Small Noise Limit of Order-Based Diffusion Processes. Electron. J. Probab. 19 (2014) 1–36. | DOI | MR | Zbl

B. Jourdain and J. Reygner, Capital Distribution and Portfolio Performance in the Mean-Field Atlas Model. Ann. Finance 11 (2015) 151–198. | DOI | MR | Zbl

I. Karatzas, S. Pal and M. Shkolnikov, Systems of Brownian Particles with Asymmetric Collisions. Ann. Inst. Henri Poincaré 52 (2016) 323–354. | DOI | MR | Zbl

I. Karatzas, A. Sarantsev, Diverse Market Models of Competing Brownian Particles with Splits and Mergers. Ann. Appl. Probab. 26 (2016) 1329–1361. | DOI | MR | Zbl

C. Kipnis, Central Limit Theorems for Infinite Series of Queues and Applications to Simple Exclusion Processes. Stoch. Proc. Appl. 14 (1986) 397–408. | MR | Zbl

C. Landim and S.B. Volchan, Equilibrium Fluctuations for a Driven Tracer Particle Dynamics. Stoch. Proc. Appl. 85 (2000) 139–158. | DOI | MR | Zbl

N. O’Connell and M. Yor, Brownian Analogues of Burke’s Theorem. Stoch. Proc. Appl. 96 (2001) 285–304. | DOI | MR | Zbl

S. Pal, Concentration for Multidimensional Diffusions and their Boundary Local Times. Probab. Theory Relat. Fields 154 (2012) 225–254. | DOI | MR | Zbl

S. Pal and J. Pitman, One-Dimensional Brownian Particle Systems with Rank-Dependent Drifts. Ann. Appl. Probab. 18 (2008) 2179–2207. | MR | Zbl

S. Pal and M. Shkolnikov, Concentration of Measure for Brownian Particle Systems Interacting Through Their Ranks. Ann. Appl. Probab. 24 (2014) 1482–1508. | MR | Zbl

V. Pipiras and M.S. Taqqu, Long-Range Dependence and Self-Similarity. Cambridge University Press (2017). | MR

A. Ruzmaikina and M. Aizenman, Characterization of Invariant Measures at the Leading Edge for Competing Particle Systems. Ann. Probab. 33 (2005) 82–113. | DOI | MR | Zbl

J. Reygner, Chaoticity of the Stationary Distribution of Rank-Based Interacting Diffusions. Electron. Commun. Probab. 20 (2015) 1–20. | DOI | MR | Zbl

A. Sarantsev, Comparison Techniques for Competing Brownian Particles. Preprint (2015). To appear in J. Theory Probab. (2018). | arXiv | MR

A. Sarantsev, Triple and Simultaneous Collisions of Competing Brownian Particles. Electron. J. Probab. 20 (2015) 1–28. | DOI | MR | Zbl

A. Sarantsev, Infinite Systems of Competing Brownian Particles. Ann. Inst. Henri Poincaré 53 (2017) 2279–2315. | DOI | MR | Zbl

A. Sarantsev, Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution. J. Theoret. Probab. 30 (2017) 1200–1223. | DOI | MR | Zbl

A. Sarantsev, Explicit Rates of Exponential Convergence for Reflected Jump-Diffusions on the Half-Line. ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016) 1069–1093. | DOI | MR | Zbl

A. Sarantsev, Stable Systems of Competing Levy Particles. Preprint (2016). | arXiv

A. Sarantsev and Li-Cheng Tsai, Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles. Electron. J. Probab. 22 (2017) 1–20. | DOI | MR | Zbl

T. Sasamoto and H. Spohn, Point-Interacting Brownian Motions in the KPZ Universality Class. Electron. J. Probab. 20 (2015) 1–28. | MR | Zbl

T. Seppalainen, A Scaling Limit for Queues in Series. Ann. Appl. Probab. 7 (1997) 885–872. | DOI | MR | Zbl

M. Shkolnikov, Competing Particle Systems Evolving by Interacting Lévy Processes. Ann. Appl. Probab. 21 (2011)1911–1932. | DOI | MR | Zbl

M. Shkolnikov, Large Systems of Diffusions Interacting Through Their Ranks. Stoch. Proc. Appl. 122 (2012) 1730–1747. | DOI | MR | Zbl

F. Spitzer, Interaction of Markov Processes. Adv. Math. 5 (1970) 246–290. | DOI | MR | Zbl

A.-S. Sznitman, A Propagation of Chaos Result for Burgers Equation. IMA Vol. Math. Appl. 9 (1986) 181–188. | MR | Zbl

A.-S. Sznitman, Topics in Propagation of Chaos. Lect. Notes Math. 1464 165–251. Springer (1991). | MR | Zbl

L.-Ch. Tsai, Infinite Dimensional Stochastic Differential Equations for Dyson’s Model. Probab. Theory Relat. Fields. 166 (2015) 801–850. | DOI | MR | Zbl

L.-Cheng Tsai, Stationary Distributions of the Atlas Model. Preprint (2017). | arXiv | MR

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