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 Electronic Communications in Probability > Vol. 15(2010) > Paper 18 open journal systems 


Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

Tomasz Bojdecki, Institute of Mathematics, University of Warsaw
Luis G. Gorostiza, Centro de Investigacion y de Estudios Avanzados, Mexico
Anna Talarczyk, Institute of Mathematics, University of Warsaw


Abstract
We consider particle systems in R with initial configurations belonging to a class of measures that obey a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z). The particles move independently according to an alpha-stable Levy process, alpha>1, and we also consider the model where they undergo critical branching. Occupation time fluctuation limits of such systems have been studied in the Poisson case. For the branching system in ``low'' dimension the limit was characterized by a process called sub-fractional Brownian motion, and this process was attributed to the branching because it had appeared only in that case. In the present more general framework sub-fractional Brownian motion is more prevalent, namely, it also appears as a component of the limit for the system without branching in ``low'' dimension. A new method of proof, based on the central limit theorem, is used.


Full text: PDF

Pages: 191-202

Published on: June 8, 2010
Bibliography
  1. Bardina, X.; Bascompte, D. A decomposition and weak approximation of the sub-fractional Brownian motion, arXiv: PR0905.4360 (2009).
  2. Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. Sub-fractional Brownian motion and its relation to occupation times. Statist. Probab. Lett. 69 (2004), no. 4, 405--419. MR2091760 (2005k:60124)
  3. Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A. Limit theorems for occupation time fluctuations of branching systems. I. Long-range dependence. Stochastic Process. Appl. 116 (2006), no. 1, 1--18. MR2186101 (2007b:60083)
  4. Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A. Limit theorems for occupation time fluctuations of branching systems. II. Critical and large dimensions. Stochastic Process. Appl. 116 (2006), no. 1, 19--35. MR2186102 (2007b:60084)
  5. Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. Self-similar stable processes arising from high-density limits of occupation times of particle systems. Potential Anal. 28 (2008), no. 1, 71--103. MR2366400 (2009c:60119)
  6. Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses. Electron. J. Probab. 14 (2009), no. 46, 1328--1371. MR2511286
  7. Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. Particle systems with quasi-homogeneous initial states and their occupation time fluctuations, arXiv: PR1002.4152 (2010).
  8. Hambly, B. M.; Jones, L. A. Erratum to ``Number variance from a probabilistic perspective, infinite systems of independent Brownian motions and symmetric $alpha$-stable processes'' [ MR2318413]. Electron. J. Probab. 14 (2009), No. 37, 1074--1079. MR2506125 (Review)
  9. Lei, Pedro; Nualart, David. A decomposition of the bifractional Brownian motion and some applications. Statist. Probab. Lett. 79 (2009), no. 5, 619--624. MR2499385 (Review)
  10. Miƚoś, Piotr. Occupation time fluctuations of Poisson and equilibrium finite variance branching systems. Probab. Math. Statist. 27 (2007), no. 2, 181--203. MR2445992 (2009i:60079)
  11. Miƚoś, Piotr. Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions. Probab. Math. Statist. 28 (2008), no. 2, 235--256. MR2548971
  12. Miƚoś, Piotr. Occupation time fluctuation limits of infinite variance equilibrium branching systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), no. 4, 593--612. MR2590158
  13. Ruiz de Chávez, J.; Tudor, C. A decomposition of sub-fractional Brownian motion. Math. Rep. (Bucur.) 11(61) (2009), no. 1, 67--74. MR2506510 (2010c:60128)
  14. Stone, Charles. On a theorem by Dobrushin. Ann. Math. Statist 39 1968 1391--1401. MR0231441 (37 #6996)
  15. Tudor, Constantin. Some properties of the sub-fractional Brownian motion. Stochastics 79 (2007), no. 5, 431--448. MR2356519 (2008i:60070)
  16. Tudor, Constantin. Inner product spaces of integrands associated to subfractional Brownian motion. Statist. Probab. Lett. 78 (2008), no. 14, 2201--2209. MR2458028 (2010b:60161)
  17. Tudor, Constantin. On the Wiener integral with respect to a sub-fractional Brownian motion on an interval. J. Math. Anal. Appl. 351 (2009), no. 1, 456--468. MR2472957 (2010d:60127)
  18. Tudor, Constantin. Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion (preprint).
  19. Yan, L.; Shen, G. On the collision local time of sub-fractional Brownian motions, Stat. Prob. Lett. 80 (2010), 296-308.
















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Electronic Communications in Probability. ISSN: 1083-589X