Construction of Immigration Superprocesses with Dependent Spatial Motion from One-Dimensional Excursions

Central limit theorems for supercritical superprocesses with immigration

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren, Song and Zhang[2015]. We first give law of large numbers for supercritical superprocesses with immigration since there is few convergence result on immigration superprocesses, then based on these results, we establish the central limit theorem.

Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

A non-critical branching immigration superprocess with dependent spatial motion is constructed and characterized as the solution of a stochastic equation driven by a time-space white noise and an orthogonal martingale measure. A representation of its conditional log-Laplace functionals is established, which gives the uniqueness of the solution and hence its Markov property. Some properties of the superprocess including an ergodic theorem are also obtained.

Path-valued branching processes and nonlocal branching superprocesses

A family of continuous-state branching processes with immigration are constructed as the solution flow of a stochastic equation system driven by time-space noises. The family can be regarded as an inhomogeneous increasing path-valued branching process with immigration. Two nonlocal branching immigration superprocesses can be defined from the flow. We identify explicitly the branching and immigration mechanisms of those processes. The results provide new perspectives into the tree-valued Markov processes of Aldous and Pitman [Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 637-686] and Abraha…

带移民与不带移民的（次）临界非局部空间分支过程的稳定性

Stability of (sub)critical non-local spatial branching processes with and without immigration

本文研究带移民与不带移民的临界及次临界非局部空间分支过程，在二阶矩条件下证明生存概率与Yaglom极限，并给出移民情形下收敛到Gamma分布及稳定性的积分判别条件。

Superprocesses with Coalescing Brownian Spatial Motion as Large-Scale Limits

A superprocess with coalescing spatial motion is constructed in terms of one-dimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the superprocess with dependent spatial motion studied in Dawson {\it et al.} (2001) and Wang (1997, 1998).

Ergodicities and exponential ergodicities of Dawson-Watanabe type processes

Under natural assumptions, we prove the ergodicities and exponential ergodicities in Wasserstein and total variation distances of Dawson--Watanabe superprocesses without or with immigration. The strong Feller property in the total variation distance is derived as a by-product. The key of the approach is a set of estimates for the variations of the transition probabilities. The estimates in Wasserstein distance are derived from an upper bound of the kernels induced by the first moment of the superprocess. Those in total variation distance are based on a comparison of the cumulant semigroup of t…