TY - JOUR
T1 - Skeleton Decomposition and Law of Large Numbers for Supercritical Superprocesses
AU - Chen, Zhen Qing
AU - Ren, Yan Xia
AU - Yang, Ting
N1 - Publisher Copyright:
© 2018, Springer Nature B.V.
PY - 2019/2/15
Y1 - 2019/2/15
N2 - 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 ψβ(x, λ) = − β(x) λ+ α(x) λ2+ ∫ (0 , ∞)(e− λ y− 1 + λy) π(x, dy) with β∈ Bb(E) , α∈Bb+(E) and π being a kernel from E to (0 , ∞) satisfying sup x ∈ E∫ (0 , ∞)(y∧ y2) π(x, dy) < ∞. 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.
AB - 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 ψβ(x, λ) = − β(x) λ+ α(x) λ2+ ∫ (0 , ∞)(e− λ y− 1 + λy) π(x, dy) with β∈ Bb(E) , α∈Bb+(E) and π being a kernel from E to (0 , ∞) satisfying sup x ∈ E∫ (0 , ∞)(y∧ y2) π(x, dy) < ∞. 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.
KW - Law of large numbers
KW - Skeleton decomposition
KW - Spectral gap
KW - Superprocesses
KW - h-Transform
UR - http://www.scopus.com/inward/record.url?scp=85047658906&partnerID=8YFLogxK
U2 - 10.1007/s10440-018-0190-1
DO - 10.1007/s10440-018-0190-1
M3 - Article
AN - SCOPUS:85047658906
SN - 0167-8019
VL - 159
SP - 225
EP - 285
JO - Acta Applicandae Mathematicae
JF - Acta Applicandae Mathematicae
IS - 1
ER -