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arXiv 1903.03751math.PR
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超Ornstein-Uhlenbeck过程的稳定中心极限定理

Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Yan-Xia Ren, Renming Song, Zhenyao Sun, Jianjie Zhao

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摘要：

本文研究了超临界$(ξ,ψ)$ -超进程$(X_t)_{t\geq 0}$的渐近行为，其基本空间运动$ξ$是$\mathbb R^d$上的Ornstein-Uhlenbeck过程，生成元为$L = \frac{1}{2}σ^2Δ- b x \cdot \nabla$，其中$σ, b >0$;其分支机制$ψ$满足Grey条件和某种扰动条件，该扰动条件保证当$z\to 0$，$ψ(z)=-αz + ηz^{1+β} (1+o(1))$，$α> 0$，$η>0$和$β\in (0, 1)$ .在假定函数$f$为多项式增长的情况下，建立了$(X_t(f) )_{t\geq 0}$的一些大数定律和$(1+β)$稳定中心极限定理.中心极限定理的形式在对应于分支率相对较小、较大或临界于平衡值的三种不同状态下是不同的，在这个意义上，中心极限定理出现相变。

英文摘要：

In this paper, we study the asymptotic behavior of a supercritical $(ξ,ψ)$-superprocess $(X_t)_{t\geq 0}$ whose underlying spatial motion $ξ$ is an Ornstein-Uhlenbeck process on $\mathbb R^d$ with generator $L = \frac{1}{2}σ^2Δ- b x \cdot \nabla$ where $σ, b >0$; and whose branching mechanism $ψ$ satisfies Grey's condition and some perturbation condition which guarantees that, when $z\to 0$, $ψ(z)=-αz + ηz^{1+β} (1+o(1))$ with $α> 0$, $η>0$ and $β\in (0, 1)$. Some law of large numbers and $(1+β)$-stable central limit theorems are established for $(X_t(f) )_{t\geq 0}$, where the function $f$ is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding the branching rate being relatively small, large or critical at a balanced value.

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