## Electronic Journal of Probability

### Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses

#### Abstract

The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles moving in $\mathbb{R}^d$ according to a symmetric $\alpha$-stable L\'evy process $(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law $(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneous case), and finite measures $(\gamma > d)$. In some cases $H_T\equiv 1$ and in others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,\alpha,\beta,\gamma$. There are two thresholds for the dimension $d$. The first one, $d=\alpha/\beta+\gamma$, determines the need for high density or not in order to obtain non-trivial limits, and its relation with a.s. local extinction of the system is discussed. The second one, $d=[\alpha(2+\beta)-\gamma\vee \alpha]/\beta$\ (if $\gamma < d$), interpolates between the two extreme cases, and it is a critical dimension which separates different qualitative behaviors of the limit processes, in particular long-range dependence in low'' dimensions, and independent increments in high'' dimensions. In low dimensions the temporal part of the limit process is a new self-similar stable process which has two different long-range dependence regimes depending on relationships among the parameters. Related results for the corresponding $(d,\alpha,\beta,\gamma)$-superprocess are also given.

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 46, 1328-1371.

Dates
Accepted: 15 June 2009
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819507

Digital Object Identifier
doi:10.1214/EJP.v14-665

Mathematical Reviews number (MathSciNet)
MR2511286

Zentralblatt MATH identifier
1190.60080

Rights

#### Citation

Bojdecki, Tomasz; Gorostiza, Luis; Talarczyk, Anna. Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses. Electron. J. Probab. 14 (2009), paper no. 46, 1328--1371. doi:10.1214/EJP.v14-665. https://projecteuclid.org/euclid.ejp/1464819507

#### References

• Billingsley, Patrick. Convergence of probability measures.John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
• Birkner, Matthias; Zähle, Iljana. A functional CLT for the occupation time of a state-dependent branching random walk. Ann. Probab. 35 (2007), no. 6, 2063–2090.
• Bojdecki, T.; Gorostiza, L. G.; Ramaswamy, S. Convergence of ${scr S}'$-valued processes and space-time random fields. J. Funct. Anal. 66 (1986), no. 1, 21–41.
• Bojdecki. T, Gorostiza L.G. and Talarczyk. A, Sub-fractional Brownian motion and its relation to occupation times, Stat. Prob. Lett. 69, (2004), 405-419.
• 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.
• 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.
• Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. A long range dependence stable process and an infinite variance branching system. Ann. Probab. 35 (2007), no. 2, 500–527.
• Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. Occupation time fluctuations of an infinite-variance branching system in large dimensions. Bernoulli 13 (2007), no. 1, 20–39.
• Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A. Occupation time limits of inhomogeneous Poisson systems of independent particles. Stochastic Process. Appl. 118 (2008), no. 1, 28–52.
• 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.
• Breiman, Leo. Probability.Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont. 1968 ix+421 pp.
• Cohen, Serge; Samorodnitsky, Gennady. Random rewards, fractional Brownian local times and stable self-similar processes. Ann. Appl. Probab. 16 (2006), no. 3, 1432–1461.
• Cox, J. Theodore; Griffeath, David. Occupation times for critical branching Brownian motions. Ann. Probab. 13 (1985), no. 4, 1108–1132.
• 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.
• Dawson, D. A.; Gorostiza, L. G.; Wakolbinger, A. Occupation time fluctuations in branching systems. J. Theoret. Probab. 14 (2001), no. 3, 729–796.
• Dawson, Donald A.; Perkins, Edwin A. Historical processes. Mem. Amer. Math. Soc. 93 (1991), no. 454, iv+179 pp.
• Dawson, Donald A.; Perkins, Edwin A. Measure-valued processes and renormalization of branching particle systems. Stochastic partial differential equations: six perspectives, 45–106, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999.
• Deuschel, Jean-Dominique; Rosen, Jay. Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes. Ann. Probab. 26 (1998), no. 2, 602–643.
• Dong, Zhao; Feng, Shui. Occupation time processes of super-Brownian motion with cut-off branching. J. Appl. Probab. 41 (2004), no. 4, 984–997.
• Theory and applications of long-range dependence. Edited by Paul Doukhan, George Oppenheim and Murad S. Taqqu.Birkhäuser Boston, Inc., Boston, MA, 2003. xii+719 pp. ISBN: 0-8176-4168-8
• Engländer, János; Kyprianou, Andreas E. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2004), no. 1A, 78–99.
• 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
• Fleischmann, Klaus; Gärtner, Jürgen. Occupation time processes at a critical point. Math. Nachr. 125 (1986), 275–290.
• Vakolbinger, A.; Vatutin, V. A.; Fleishmann, K. Branching systems with long-lived particles at a critical dimension.(Russian) Teor. Veroyatnost. i Primenen. 47 (2002), no. 3, 417–451; translation in Theory Probab. Appl. 47 (2003), no. 3, 429–454
• Gorostiza, L. G.; López-Mimbela, J. A. An occupation time approach for convergence of measure-valued processes, and the death process of a branching system. Statist. Probab. Lett. 21 (1994), no. 1, 59–67.
• Gorostiza, Luis G.; Navarro, Reyla; Rodrigues, Eliane R. Some long-range dependence processes arising from fluctuations of particle systems. Acta Appl. Math. 86 (2005), no. 3, 285–308.
• Gorostiza, Luis G.; Wakolbinger, Anton. Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab. 19 (1991), no. 1, 266–288.
• Heyde, C. C.; Yang, Y. On defining long-range dependence. J. Appl. Probab. 34 (1997), no. 4, 939–944.
• Heyde, C. C. On modes of long-range dependence. J. Appl. Probab. 39 (2002), no. 4, 882–888.
• Hambly, Ben; Jones, Liza. Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric $alpha$-stable processes. Electron. J. Probab. 12 (2007), no. 30, 862–887 (electronic).
• Hong, Wenming. Longtime behavior for the occupation time process of a super-Brownian motion with random immigration. Stochastic Process. Appl. 102 (2002), no. 1, 43–62.
• Houdré, Christian; Villa, José. An example of infinite dimensional quasi-helix. Stochastic models (Mexico City, 2002), 195–201, Contemp. Math., 336, Amer. Math. Soc., Providence, RI, 2003.
• Iscoe, I. A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71 (1986), no. 1, 85–116.
• Iscoe, I. On the supports of measure-valued critical branching Brownian motion. Ann. Probab. 16 (1988), no. 1, 200–221.
• Iscoe, Ian; Lee, Tzong-Yow. Large deviations for occupation times of measure-valued branching Brownian motions. Stochastics Stochastics Rep. 45 (1993), no. 3-4, 177–209.
• Kaj I. and Taqqu M. S, Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In and Out of Equilibrium 2. Eds. M. E. Vares, V. Sidoravicius. Progress in Probability, Vol. 60, 383-427. Birkhauser, 2008.
• Klenke, Achim. Clustering and invariant measures for spatial branching models with infinite variance. Ann. Probab. 26 (1998), no. 3, 1057–1087.
• Lee, Tzong-Yow; Remillard, Bruno. Large deviations for the three-dimensional super-Brownian motion. Ann. Probab. 23 (1995), no. 4, 1755–1771.
• Levy, Joshua B.; Taqqu, Murad S. Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards. Bernoulli 6 (2000), no. 1, 23–44.
• Li, Zenghu; Zhou, Xiaowen. Distribution and propagation properties of superprocesses with general branching mechanisms. Commun. Stoch. Anal. 2 (2008), no. 3, 469–477.
• Maejima, Makoto; Yamamoto, Kenji. Long-memory stable Ornstein-Uhlenbeck processes. Electron. J. Probab. 8 (2003), no. 19, 18 pp. (electronic).
• Méléard, S.; Roelly, S. An ergodic result for critical spatial branching processes. Stochastic analysis and related topics (Silivri, 1990), 333–341, Progr. Probab., 31, Birkhäuser Boston, Boston, MA, 1992.
• Miłoś›, Piotr. Occupation time fluctuations of Poisson and equilibrium finite variance branching systems. Probab. Math. Statist. 27 (2007), no. 2, 181–203. (Review)
• Miłoś, P. Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions, Probab. Math. Statist. 28 (2), (2008), 235-256.
• Miłoś, P. Occupation time fluctuation limits of infinite variance equilibrium branching systems, (preprint), Math. ArXiv. PR 0802.0187.
• Mitoma, Itaru. Tightness of probabilities on $C([0,1];{cal S}sp{prime} )$ and $D([0,1];{cal S}sp{prime} )$. Ann. Probab. 11 (1983), no. 4, 989–999.
• Perkins, Edwin. Polar sets and multiple points for super-Brownian motion. Ann. Probab. 18 (1990), no. 2, 453–491.
• Perkins, Edwin. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781, Springer, Berlin, 2002.
• Pipiras, Vladas; Taqqu, Murad S.; Levy, Joshua B. Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli 10 (2004), no. 1, 121–163.
• Rosiński, Jan; Zak, Tomasz. The equivalence of ergodicity of weak mixing for infinitely divisible processes. J. Theoret. Probab. 10 (1997), no. 1, 73–86.
• Samorodnitsky, Gennady. Long range dependence. Found. Trends Stoch. Syst. 1 (2006), no. 3, 163–257. ISBN: 978-1-60198-090-8
• Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes.Stochastic models with infinite variance.Stochastic Modeling. Chapman & Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0
• Shiozawa, Yuichi. Extinction of branching symmetric $alpha$-stable processes. J. Appl. Probab. 43 (2006), no. 4, 1077–1090.
• Talarczyk, Anna. A functional ergodic theorem for the occupation time process of a branching system. Statist. Probab. Lett. 78 (2008), no. 7, 847–853.
• Taqqu, Murad S. Fractional Brownian motion and long-range dependence. Theory and applications of long-range dependence, 5–38, Birkhäuser Boston, Boston, MA, 2003.
• Vakolbinger, A.; Vatutin, V. A. Branching processes in long-lived particles.(Russian) Teor. Veroyatnost. i Primenen. 43 (1998), no. 4, 655–671; translation in Theory Probab. Appl. 43 (1999), no. 4, 620–632
• Zhang, Mei. Functional central limit theorem for the super-Brownian motion with super-Brownian immigration. J. Theoret. Probab. 18 (2005), no. 3, 665–685.
• Zhou, Xiaowen. A zero-one law of almost sure local extinction for $(1+beta)$-super-Brownian motion. Stochastic Process. Appl. 118 (2008), no. 11, 1982–1996.