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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
Article
  • 33 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

60J68 60F15 60F25 

Notes

Acknowledgements

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.

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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

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