[论文解读] Continuous-state branching processes with immigration

Zenghu Li|arXiv (Cornell University)|Jan 11, 2019

Stochastic processes and statistical mechanics被引用 5

一句话总结

本文为连续状态分支过程带移民（CBI-processes）提供了简洁且易懂的介绍，聚焦于随机方程、路径结构及长期行为。在较弱条件下建立了强Feller性质与指数遍历性，通过耦合技术证明了以指数速率收敛至唯一平稳分布。

ABSTRACT

This work provides a brief introduction to continuous-state branching processes (CB-processes) and continuous-state branching processes with immigration (CBI-processes) accessible to graduate students with reasonable background in probability theory and stochastic processes. In particular, we give a quick development of the stochastic equations of the processes and some immediate applications. The proofs given here are more elementary than those appearing in the literature before. We have made them readable without requiring too much preliminary knowledge on branching processes and stochastic analysis. In Section 1, we review some properties of Laplace transforms of finite measures on the positive half line. In Section 2, a construction of CB-processes is given as rescaling limits of Galton--Watson branching processes. This approach also gives the physical interpretation of the CB-processes. Some basic properties of the processes are developed in Section 3. The Laplace transforms of some positive integral functionals are calculated explicitly in Section 4. In Section 5, the CBI-processes are constructed as rescaling limits of Galton--Watson branching processes with immigration. In Section 6, we present reconstructions of the CB- and CBI-processes by Poisson random measures determined by entrance laws, which reveal the structures of the trajectories of the processes. Several equivalent formulations of martingale problems for CBI-processes are given in Section 7. From those we derive the stochastic equations of the processes in Section 8. Using the stochastic equations, some characterizations of local and global maximal jumps of the CB- and CBI-processes are given in Section 9. In Section 10, we prove the strong Feller property and the exponential ergodicity of the CBI-process under suitable conditions using a coupling based on one of the stochastic equations.

研究动机与目标

• 为具备最少随机分析预备知识的研究生提供CBI-processes的入门介绍。
• 使用初等方法推导CBI-processes的随机方程。
• 通过耦合方法建立转移半群的强Feller性质与指数遍历性。
• 表征CBI-processes的长期行为，包括收敛至唯一平稳分布。

提出的方法

• 将CBI-processes构造为带移民的Galton–Watson分支过程的标度极限。
• 利用拉普拉斯变换与随机时间改变来刻画CB-与CBI-processes的分布。
• 从鞅问题形式化推导CBI-processes的随机方程。
• 基于强随机方程应用耦合技术，证明强Feller性质。
• 通过有界转移核与不变测度之间的总变差距离，建立指数遍历性。
• 使用函数 $ v_t(\theta) $ 分析收敛速率，并推导收敛速度的显式上界。

实验结果

研究问题

• RQ1 如何通过带移民的离散分支过程的标度极限构造CBI-processes？
• RQ2 何种条件可保证CBI-processes转移半群的强Feller性质？
• RQ3 CBI-processes的转移分布以多快的速度收敛至其平稳分布？
• RQ4 分支机制 $ \phi $ 与移民机制 $ \psi $ 在决定长期行为中起什么作用？
• RQ5 能否使用耦合方法推导出收敛至不变测度的显式收敛速率？

主要发现

• CBI-processes的转移半群 $ (P_t)_{t \geq 0} $ 满足强Feller性质，即对任意有界可测函数 $ f $，$ P_t f $ 连续。
• 在条件 $ b > 0 $ 下，CBI-processes具有唯一的平稳分布 $ \eta $，其拉普拉斯变换为 $ L_\eta(\lambda) = \exp\left\{ -\int_0^\lambda \frac{\psi(z)}{\phi(z)} dz \right\} $。
• 转移分布 $ P_t(x, \cdot) $ 收敛至 $ \eta $ 的速度为指数级，且满足上界 $ \|P_t(x, \cdot) - \eta\|_{\text{var}} \leq 2[x + b^{-1}\psi'(0)] \bar{v}_r e^{b(r-t)} $，其中 $ t \geq r > 0 $。
• 平稳分布的一阶矩为 $ \int y \eta(dy) = b^{-1} \psi'(0) $，通过在 $ \lambda = 0 $ 处对拉普拉斯变换求导得到。
• 耦合方法给出上界 $ \|P_t(x, \cdot) - P_t(y, \cdot)\|_{\text{var}} \leq 2 \bar{v}_t |x - y| $，从而确认强Feller性质。
• 收敛速率由 $ \bar{v}_t $ 控制，当 $ b > 0 $ 时其呈指数衰减，从而确保指数遍历性。