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 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.