Theory of Probability &Its Applications, Volume 66, Issue 2, Page 276-298, January 2021.
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 the superprocess with that of a continuous-state branching process. The results improve and extend considerably those of Friesen [Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration, arXiv:1903.05546, 2019] and Stannat [J. Funct. Anal., 201 (2003), pp. 185--227; Ann. Probab., 31 (2003), pp. 1377--1412]. We also show a connection between the ergodicities of the immigration superprocesses and decomposable distributions.

中文翻译：

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Dawson--Watanabe 型过程的遍历性和指数遍历性

概率论及其应用，第 66 卷，第 2 期，第 276-298 页，2021 年 1 月。
在自然假设下，我们证明了Wasserstein 中的遍历性和指数遍历性以及没有或有迁移的Dawson-Watanabe 超过程的总变异距离。总变异距离中的强 Feller 属性是作为副产品导出的。该方法的关键是对转移概率变化的一组估计。Wasserstein 距离的估计值来自于由超级过程的第一时刻引起的内核的上限。总变异距离是基于超过程的累积半群与连续状态分支过程的累积半群的比较。结果大大改进和扩展了 Friesen [Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration, arXiv:1903.05546, 2019] 和 Stannat [J. 功能。分析，201 (2003)，第 185--227 页；安。Probab., 31 (2003), pp. 1377--1412]。我们还展示了移民超级过程的遍历性与可分解分布之间的联系。