Illinois Journal of Mathematics

Limit theorems for some critical superprocesses

Yan-Xia Ren, Renming Song, and Rui Zhang

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Let $X=\{X_{t},t\ge0;\mathbb{P}_{\mu}\}$ be a critical superprocess starting from a finite measure $\mu$. Under some conditions, we first prove that $\lim_{t\to\infty}t{ \mathbb{P}}_{\mu}(\Vert X_{t}\Vert \ne0)=\nu^{-1}\langle\phi_{0},\mu\rangle$, where $\phi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator $L$ of the mean semigroup of $X$, and $\nu$ is a positive constant. Then we show that, for a large class of functions $f$, conditioning on $\Vert X_{t}\Vert \ne0$, $t^{-1}\langle f,X_{t}\rangle$ converges in distribution to $\langle f,\psi_{0}\rangle_{m}W$, where $W$ is an exponential random variable, and $\psi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the dual of $L$. Finally, if $\langle f,\psi_{0}\rangle_{m}=0$, we prove that, conditioning on $\Vert X_{t}\Vert \ne0$, $(t^{-1}\langle\phi_{0},X_{t}\rangle,t^{-1/2}\langle f,X_{t}\rangle )$ converges in distribution to $(W,G(f)\sqrt{W})$, where $G(f)\sim\mathcal{N}(0,\sigma_{f}^{2})$ is a normal random variable, and $W$ and $G(f)$ are independent.

Article information

Illinois J. Math., Volume 59, Number 1 (2015), 235-276.

Received: 17 August 2015
Revised: 16 November 2015
First available in Project Euclid: 11 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]


Ren, Yan-Xia; Song, Renming; Zhang, Rui. Limit theorems for some critical superprocesses. Illinois J. Math. 59 (2015), no. 1, 235--276. doi:10.1215/ijm/1455203166.

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  • R. Adamczak and P. Miłoś, CLT for Ornstein–Uhlenbeck branching particle system, Electron. J. Probab. 20 (2015), Art. ID 42.
  • W. J. Anderson, Continuous time Markov chains, Springer-Verlag, New York, 1991.
  • S. Asmussen and H. Hering, Branching processes, Birkhäuser, Boston, MA, 1983.
  • K. Athreya and P. Ney, Branching processes, Springer-Verlag, New York, 1972.
  • K. Athreya and P. Ney, Functionals of critical multitype branching processes, Ann. Probab. 2 (1974), 339–343.
  • R. Banuelos, Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, J. Funct. Anal. 100 (1991), 181–206.
  • N. Champagnat and S. Roelly, Limit theorems for conditioned multitype Dawson–Watanabe processes and Feller diffusions, Electron. J. Probab. 13 (2008), 777–810.
  • Z.-Q. Chen, P. Kim and R. Song, Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation, Ann. Probab. 40 (2012), 2483–2538.
  • Z.-Q. Chen, P. Kim and R. Song, Dirichlet heat kernel estimates for rotationally symmetric Levy processes, Proc. Lond. Math. Soc. (3) 109 (2014), 90–120.
  • Z.-Q. Chen, P. Kim and R. Song, Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components, J. Reine Angew. Math. 711 (2016), 111–138.
  • Z.-Q. Chen, Y.-X. Ren, R. Song and R. Zhang, Strong law of large numbers for supercritical superprocesses under second moment condition, Front. Math. China 10 (2015), 807–838.
  • E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), 335–395.
  • D. A. Dawson, Measure-valued Markov processes, Springer-Verlag, Berlin, 1993.
  • R. M. Dudley, Real analysis and probability, Cambridge University Press, Cambridge, 2002.
  • E. B. Dynkin, Superprocesses and partial differential equations, Ann. Probab. 21 (1993), 1185–1262.
  • E. B. Dynkin and S. E. Kuznetsov, $\mathbb{ N}$-measure for branching exit Markov system and their applications to differential equations, Probab. Theory Related Fields 130 (2004), 135–150.
  • N. El Karoui and S. Roelly, Propriétés de martingales, explosion et représentation de Lévy–Khintchine d'une classe de processus de branchment à valeurs mesures, Stochastic Process. Appl. 38 (1991), 239–266.
  • J. Engländer, Y.-X. Ren and R. Song, Weak extinction versus global exponential growth of total mass for superdiffusions corresponding to the operator $Lu+\beta u-ku^2$, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), 448–482.
  • S. N. Evans and E. Perkins, Measure-valued Markov branching processes conditioned on non-extinction, Israel J. Math. 71 (1990), 329–337.
  • M. I. Goldstein and F. M. Hoppe, Critical multitype branching processes with infinite variance, J. Math. Anal. Appl. 65 (1978), 675–686.
  • T. E. Harris, The theory of branching processes, Springer, Berlin, 1963.
  • A. Joffe and F. Spitzer, On multitype branching processes with $\rho\le1$, J. Math. Anal. Appl. 19 (1967), 409–430.
  • K. Kaleta and T. Kulczycki, Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians, Potential Anal. 33 (2010), 313–339.
  • O. Kallenberg, Foundations of modern probability, 2nd ed., Springer, New York, 2002.
  • H. Kesten, P. Ney and F. Spitzer, The Galton–Watson process with mean one and finite variance, Theory Probab. Appl. 11 (1966), 513–540.
  • P. Kim and R. Song, Two-sided estimates on the density of Brownian motion with singular drift, Illinois J. Math. 50 (2006), 635–688.
  • P. Kim and R. Song, On dual processes of non-symmetric diffusions with measure-valued drifts, Stochastic Process. Appl. 118 (2008), 790–817.
  • P. Kim and R. Song, Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains, Tohoku Math. J. (2) 60 (2008), 527–547.
  • P. Kim and R. Song, Intrinsic ultracontractivity of non-symmetric diffusions with measure-valued drifts and potentials, Ann. Probab. 36 (2008), 1904–1945.
  • P. Kim and R. Song, Intrinsic ultracontractivity for non-symmetric Lévy processes, Forum Math. 21 (2009), 43–66.
  • P. Kim and R. Song, Stable process with singular drift, Stochastic Process. Appl. 124 (2014), 2479–2516.
  • K. Kim, R. Song and Z. Vondracek, Two-sided Green function estimates for killed subordinate Brownian motions, Proc. Lond. Math. Soc. (3) 104 (2012), 927–958.
  • K. Kim, R. Song and Z. Vondracek, Potential theory of subordinate Brownian motions with Gaussian components, Stochastic Process. Appl. 123 (2013), 764–795.
  • A. Kolmogorov, Zur Lösung einer biologischen Aufgabe, Izvestiya nauchno-issledovatelskogo instituta matematiki i mechaniki pri Tomskom Gosudarstvennom Universitete 2 (1938), 1–6.
  • T. Kulczycki and B. Siudeja, Intrinsic ultracontractivity of the Feynman–Kac semigroup for relativistic stable processes, Trans. Amer. Math. Soc. 358 (2006), 5025–5057.
  • A. Lambert, Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct, Electron. J. Probab. 12 (2007), 420–446.
  • Z. Li, Asymptotic behavior of continuous time and state branching process, J. Austral. Math. Soc. Ser. A 68 (2000), 68–84.
  • Z. Li, Skew convolution semigroups and related immigration processes, Theory Probab. Appl. 46 (2003), 274–296.
  • Z. Li, Measure-valued branching Markov processes, Springer, Heidelberg, 2011.
  • R. Lyons, R. Pemantle and Y. Peres, Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes, Ann. Probab. 23 (1995), 1125–1138.
  • P. Miłoś, Spatial CLT for the supercritical Ornstein–Uhlenbeck superprocess, preprint, 2012; available at \arxivurlarXiv:1203.6661.
  • P. Ney, Critical multi-type degenerate branching processes, technical report, University of Wisconsin.
  • A. G. Pakes, Critical Markov branching process limit theorems allowing infinite variance, Adv. in Appl. Probab. 42 (2010), 460–488.
  • Y.-X. Ren, R. Song and R. Zhang, Central limit theorems for super Ornstein–Uhlenbeck processes, Acta Appl. Math. 130 (2014), 9–49.
  • Y.-X. Ren, R. Song and R. Zhang, Central limit theorems for supercritical branching Markov processes, J. Funct. Anal. 266 (2014), 1716–1756.
  • Y.-X. Ren, R. Song and R. Zhang, Central limit theorems for supecritical superprocesses, Stochastic Process. Appl. 125 (2015), 428–457.
  • Y.-X. Ren, R. Song and R. Zhang, Central Limit Theorems for Supercritical Branching Non-symmetric Markov Processes, to appear in Ann. Probab.; available at \arxivurlarXiv:1404.0116.
  • H. H. Schaefer, Banach lattices and positive operators, Springer, New York, 1974.
  • E. Seneta, The Galton–Watson process with mean one, J. Appl. Probab. 4 (1967), 489–495.
  • Y.-C. Sheu, Lifetime and compactness of range for super-Brownian motion with a general branching mechanism, Stochastic Process. Appl. 70 (1997), 129–141.
  • R. S. Slack, A branching process with mean one and possibly infinite variance, Z. Wahrsch. Verw. Gebiete 9 (1968), 139–145.
  • V. A. Vatutin, Limit theorems for critical multitype Markov branching processes with infinite second moments, Mat. Sb. 103 (1977), 253–264.
  • V. A. Vatutin, Linear functionals for critical multitype Galton–Watson branching processes, J. Math. Sci. (N. Y.) 99 (2000), 1502–1509.
  • A. M. Yaglom, Certain limit theorems of the theory of branching random processes, Doklady Akad. Nauk SSSR (N.S.) 56 (1947), 795–798.
  • J. Zhang, S. Li and R. Song, Quasi-stationarity and quasi-ergodicity of general Markov processes, Sci. China Math. 57 (2014), 2013–2024.