Acta Applicandae Mathematicae

, Volume 159, Issue 1, pp 225–285 | Cite as

Skeleton Decomposition and Law of Large Numbers for Supercritical Superprocesses

  • Zhen-Qing Chen
  • Yan-Xia Ren
  • Ting YangEmail author


The goal of this paper is twofold. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space \(E\) and the branching mechanism takes the form
$$ \psi _{\beta }(x,\lambda )=-\beta (x)\lambda +\alpha (x)\lambda ^{2}+ \int _{(0,{\infty })}\bigl(e^{-\lambda y}-1+\lambda y\bigr)\pi (x,dy) $$
with \(\beta \in \mathcal{B}_{b}(E)\), \(\alpha \in \mathcal{B}^{+}_{b}(E)\) and \(\pi \) being a kernel from \(E\) to \((0,{\infty })\) satisfying \(\sup_{x\in E}\int _{(0,{\infty })} (y\wedge y^{2}) \pi (x,dy)<{\infty }\). The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap. Our results cover many interesting examples of superprocesses, including super Ornstein-Uhlenbeck process and super stable-like process. The strong law of large numbers for supercritical superprocesses are obtained under the assumption that the strong law of large numbers for an associated supercritical branching Markov process holds along a discrete sequence of times, extending an earlier result of Eckhoff et al. (Ann. Probab. 43(5):2594–2659, 2015) for superdiffusions to a large class of superprocesses. The key for such a result is due to the skeleton decomposition of superprocess, which represents a superprocess as an immigration process along a supercritical branching Markov process.


Law of large numbers Superprocesses Skeleton decomposition \(h\)-Transform Spectral gap 

Mathematics Subject Classification (2010)

60J68 60F15 60F25 



We thank Zenghu Li for helpful discussions on Kuznetsov measure for superprocesses. We also thank the referees for their helpful comments on the first version of this paper.


  1. 1.
    Albeverio, S., Blanchard, P., Ma, Z.-M.: Feynman-Kac semigroups in terms of signed smooth measures. In: Hornung, U., et al. (eds.) Random Partial Differential Equations, pp. 1–31. Birkhäuser, Basel (1991) Google Scholar
  2. 2.
    Berestycki, J., Kyprianou, A.E., Murillo-Salas, A.: The prolific backbone for supercritical superprocesses. Stoch. Process. Appl. 121, 1315–1331 (2011) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Carmona, R., Masters, W.C., Simon, B.: Relativistic Schrödinger operators: asymptotiic behavior of the eigenfunctions. J. Funct. Anal. 91, 117–142 (1990) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, Z.-Q., Fitzsimmons, P.J., Takeda, M., Ying, J., Zhang, T.-S.: Absolute continuity of symmetric Markov processes. Ann. Probab. 32, 2067–2098 (2004) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Changes, and Boundary Theory. Princeton University Press, Princeton (2012) zbMATHGoogle Scholar
  6. 6.
    Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on \(d\)-sets. Stoch. Process. Appl. 108, 27–62 (2003) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, Z.-Q., Ren, Y.-X., Song, R., Zhang, R.: Strong law of large numbers for supercritical superprocesses under second moment condition. Front. Math. China 10(4), 807–838 (2015) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen, Z.-Q., Ren, Y.-X., Wang, H.: An almost sure scaling limit theorem for Dawson-Watanabe superprocesses. J. Funct. Anal. 254, 1988–2019 (2008) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chen, Z.-Q., Ren, Y.-X., Yang, T.: Law of large numbers for branching symmetric Hunt processes with measure-valued branching rates. J. Theor. Probab. 30(3), 898–931 (2017) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Durrett, R.: Probability Theory and Examples, 4nd edn. Cambridge University Press, Cambridge (2010) zbMATHGoogle Scholar
  11. 11.
    Dynkin, E.B.: Superprocesses and partial differential equations. Ann. Probab. 21(3), 1185–1262 (1993) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dynkin, E.B.: Branching exit Markov systems and superprocesses. Ann. Probab. 29(4), 1833–1858 (2001) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dynkin, E.B.: Diffusions, Superdiffusions and Partial Differential Equations. Amer. Math. Soci, Providence (2002) zbMATHGoogle Scholar
  14. 14.
    Dynkin, E.B., Kuznetsov, S.E.: ℕ-Measures for branching exit Markov systems and their apllications to differential equations. Probab. Theory Relat. Fields 130(1), 135–150 (2004) zbMATHGoogle Scholar
  15. 15.
    Eckhoff, M., Kyprianou, A.E., Winkel, M.: Spine, skeletons and the strong law of large numbers. Ann. Probab. 43(5), 2594–2659 (2015) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Engländer, J.: Law of large numbers for superdiffusions: the non-ergodic case. Ann. Inst. Henri Poincaré Probab. Stat. 45, 1–6 (2009) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Engländer, J., Pinsky, R.G.: On the construction and support properties of measure-valued diffusions on \(D\subseteq \mathbb{R}^{d}\) with spatially dependent branching. Ann. Probab. 27(2), 684–730 (1999) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Engländer, J., Kyprianou, A.E.: Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32(1A), 78–99 (2003) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Engländer, J., Turaev, D.: A scaling limit theorem for a class of superdiffusions. Ann. Probab. 30(2), 683–722 (2002) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Engländer, J., Winter, A.: Law of large numbers for a class of superdiffusions. Ann. Inst. Henri Poincaré Probab. Stat. 42(2), 171–185 (2006) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin (1994) zbMATHGoogle Scholar
  22. 22.
    Kim, P., Song, R.: Two-sided estimates on the density of Brownian motion with singular drift. Ill. J. Math. 50, 635–688 (2006) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kouritzin, M.A., Ren, Y.-X.: A strong law of large numbers for super-stable processes. Stoch. Process. Appl. 124(1), 505–521 (2014) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kyprianou, A.E., Liu, R.-L., Murillo-Salas, A., Ren, Y.-X.: Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Ann. Inst. Henri Poincaré Probab. Stat. 48(3), 661–687 (2012) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kyprianou, A.E., Perez, J.-L., Ren, Y.-X.: The backbone decomposition for a general spatially dependent supercritical superprocesses. In: Lecture Notes in Mathematics, Séminaire de Probabilités, vol. 2123, (2014) 46, 33–59 Google Scholar
  26. 26.
    Li, Z.-H.: Measure-Valued Branching Markov Processes. Springer, Heidelberg (2011) zbMATHGoogle Scholar
  27. 27.
    Liu, R.-L., Ren, Y.-X., Song, R.: Strong law of large numbers for a class of superdiffusions. Acta Appl. Math. 123(1), 73–97 (2013) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) zbMATHGoogle Scholar
  29. 29.
    Ren, Y.-X., Song, R., Yang, T.: Spine decomposition and \(L\log L\) criterion for superprocesses with non-local branching mechanisms (2016).
  30. 30.
    Shiozawa, Y.: Exponential growth of the numbers of particles for branching symmetric \(\alpha \)-stable processes. J. Math. Soc. Jpn. 60, 75–116 (2008) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Takeda, M.: Large deviations for additive functionals of symmetirc stable processes. J. Theor. Probab. 21(2), 336–355 (2008) zbMATHGoogle Scholar
  32. 32.
    Wang, L.: An almost sure limit theorem for super-Brownian motion. J. Theor. Probab. 23(2), 401–416 (2010) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.LMAM School of Mathematical Sciences & Center for Statistical SciencePeking UniversityBeijingP.R. China
  3. 3.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingP.R. China
  4. 4.Beijing Key Laboratory on MCAACIBeijing Institute of TechnologyBeijingP.R. China

Personalised recommendations