## Electronic Journal of Probability

### CLT for Ornstein-Uhlenbeck branching particle system

#### Abstract

In this paper we consider a branching particle system consisting of particles moving according to an Ornstein-Uhlenbeck process in $\mathbb{R}d$ and undergoing binary, supercritical branching with a constant rate $\lambda > 0$. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. The limit and the CLT normalization fall into three qualitatively different classes. In what we call the small branching rate case the situation resembles the classical one. The weak limit is Gaussian and normalization is the square root of the size of the system. In the critical case the limit is still Gaussian, but the normalization requires an additional term. Finally, when branching has a large rate the situation is completely different. The limit is no longer Gaussian, the normalization is substantially larger than the classical one and the convergence holds in probability.  We also prove that the spatial fluctuations are asymptotically independent of the fluctuations of the total number of particles (which is a Galton-Watson process).

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 42, 35 pp.

Dates
Accepted: 12 April 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067148

Digital Object Identifier
doi:10.1214/EJP.v20-4233

Mathematical Reviews number (MathSciNet)
MR3339862

Zentralblatt MATH identifier
1321.60035

Rights

#### Citation

Adamczak, Radosław; Miłoś, Piotr. CLT for Ornstein-Uhlenbeck branching particle system. Electron. J. Probab. 20 (2015), paper no. 42, 35 pp. doi:10.1214/EJP.v20-4233. https://projecteuclid.org/euclid.ejp/1465067148

#### References

• Adamczak, Radosław; Miłoś, Piotr. $U$-statistics of Ornstein-Uhlenbeck branching particle system. J. Theoret. Probab. 27 (2014), no. 4, 1071–1111.
• Cécile Ané, Sébastien Blachère, Djalil Chafaï, Pierre Fougères, Ivan Gentil, Florent Malrieu, Cyril Roberto, and Grégory Scheffer, Sur les inégalités de Sobolev logarithmiques, Panoramas et SynthÃ¨ses [Panoramas and Syntheses], vol. 10, Société Mathématique de France, Paris, 2000, With a preface by Dominique Bakry and Michel Ledoux.
• Asmussen, Soren; Hering, Heinrich. Strong limit theorems for general supercritical branching processes with applications to branching diffusions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 3, 195–212.
• Asmussen, Soren; Hering, Heinrich. Strong limit theorems for supercritical immigration-branching processes. Math. Scand. 39 (1976), no. 2, 327–342 (1977).
• Athreya, Krishna B.; Ney, Peter E. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
• Bansaye, Vincent; Delmas, Jean-François; Marsalle, Laurence; Tran, Viet Chi. Limit theorems for Markov processes indexed by continuous time Galton-Watson trees. Ann. Appl. Probab. 21 (2011), no. 6, 2263–2314.
• Biggins, J. D.; Kyprianou, A. E. Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 (1997), no. 1, 337–360.
• Da Prato, Giuseppe. An introduction to infinite-dimensional analysis. Revised and extended from the 2001 original by Da Prato. Universitext. Springer-Verlag, Berlin, 2006. x+209 pp. ISBN: 978-3-540-29020-9; 3-540-29020-6
• Dawson, Donald A. Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI - 1991, 1–260, Lecture Notes in Math., 1541, Springer, Berlin, 1993.
• Dudley, R. M. Real analysis and probability. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. x+555 pp. ISBN: 0-521-00754-2
• Dynkin, E. B. Branching particle systems and superprocesses. Ann. Probab. 19 (1991), no. 3, 1157–1194.
• Dynkin, Eugene B. An introduction to branching measure-valued processes. CRM Monograph Series, 6. American Mathematical Society, Providence, RI, 1994. x+134 pp. ISBN: 0-8218-0269-0
• Englander, Janos; Harris, Simon C.; Kyprianou, Andreas E. Strong law of large numbers for branching diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 279–298.
• Etheridge, Alison M. An introduction to superprocesses. University Lecture Series, 20. American Mathematical Society, Providence, RI, 2000. xii+187 pp. ISBN: 0-8218-2706-5
• Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
• Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
• Le Gall, Jean-François. Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1999. x+163 pp. ISBN: 3-7643-6126-3
• Graversen, S. E.; Peskir, G. Maximal inequalities for the Ornstein-Uhlenbeck process. Proc. Amer. Math. Soc. 128 (2000), no. 10, 3035–3041.
• Harris, Simon C. Convergence of a "Gibbs-Boltzmann" random measure for a typed branching diffusion. Séminaire de Probabilités, XXXIV, 239–256, Lecture Notes in Math., 1729, Springer, Berlin, 2000.
• Kallianpur, Gopinath; Xiong, Jie. Stochastic differential equations in infinite-dimensional spaces. Expanded version of the lectures delivered as part of the 1993 Barrett Lectures at the University of Tennessee, Knoxville, TN, March 25-27, 1993. With a foreword by Balram S. Rajput and Jan Rosinski. Institute of Mathematical Statistics Lecture Notes-Monograph Series, 26. Institute of Mathematical Statistics, Hayward, CA, 1995. vi+342 pp. ISBN: 0-940600-38-2
• Piotr Milo's, CLT for Ornstein-Uhlenbeck superprocess, submitted to J. Th. Probab. ARXIV1203.6661
• Mitoma, Itaru. Tightness of probabilities on $C([0,1];{\cal S}^{\prime} )$ and $D([0,1];{\cal S}^{\prime} )$. Ann. Probab. 11 (1983), no. 4, 989–999.
• Oksendal, Bernt. Stochastic differential equations. An introduction with applications. Fifth edition. Universitext. Springer-Verlag, Berlin, 1998. xx+324 pp. ISBN: 3-540-63720-6
• Ren, Yan-Xia; Song, Renming; Zhang, Rui. Central limit theorems for supercritical branching Markov processes. J. Funct. Anal. 266 (2014), no. 3, 1716–1756.
• Ren, Yan-Xia; Song, Renming; Zhang, Rui. Central limit theorems for supercritical superprocesses. Stochastic Process. Appl. 125 (2015), no. 2, 428–457.
• Rudin, Walter. Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-DÃ¼sseldorf-Johannesburg, 1973. xiii+397 pp.