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JOURNALS // Teoriya Veroyatnostei i ee Primeneniya // Archive

Teor. Veroyatnost. i Primenen., 2021 Volume 66, Issue 2, Pages 342–368 (Mi tvp5341)

This article is cited in 7 papers

Ergodicities and exponential ergodicities of Dawson–Watanabe type processes

Z. Li

Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, People's Republic of China

Abstract: 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.

Keywords: Dawson–Watanabe superprocess, immigration, coupling, strong Feller property, stationary distribution, exponential ergodicity.

Received: 08.08.2019

DOI: 10.4213/tvp5341


 English version:
Theory of Probability and its Applications, 2021, 66:2, 276–298

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© Steklov Math. Inst. of RAS, 2024