Fleming-Viot processes incorporating mutation and selection are considered. It is well-known that if the mutation factor is of uniform type, the processes have a reversible stationary distribution, and it has been an open problem to characterize the class of the processes that have a reversible stationary distributions. This paper proves that if the Fleming-Viot process has a reversible stationary distribution, then the associated mutation operator is of uniform type.

The immigration structure associated with a measure-valued branching process may be described by a skew convolution semigroup. For a special type of measure-valued branching process, the Dawson-Watanabe superprocess, we show that a skew convolution semigroup corresponds uniquely to an infinitely divisible probability measure on the space of entrance laws for the underlying process. An immigration process associated with a Borel right superprocess does not always have a right continuous realization, but it can always be obtained by transformation from a Borel right one in an enlarged state space.

Starting from the cumulant semigroup of a measure-valued branching process, we construct the transition probabilities of some Markov process Y (β) = (Yt (β), t ∈ R), which we call a measure-valued branching process with discrete immigration of unitβ. The immigration of Y (β) is governed by a Poisson random measure р on the time-distribution space and a probability generating function h; both depending on β. It is shown that, under suitable hypotheses, Y (β) approximates to a Markov process Y = (Yt, t ∈ R) as β→ 0+. The latter is the one we call a measure-valued branching process with immigration. The convergence of branching particle systems with immigration is also studied.

The scaling limit for a class of interacting superprocesses and the associated singular, degenerate stochastic partial di erential equation (SDSPDE) are investigated. It is proved that the scaling limit is a coalescing, purely-atomic-measurevalued process which is the unique strong solution of a reconstructed, associated SDSPDE.

A special type of immigration associated with measure-valued branching processes is formulated by using skew convolution semigroups. We give characterization for a general inhomogeneous skew convolution semigroup in terms of probability entrance laws. The related immigration process is constructed by summing up measure-valued paths in the Kuznetsov process determined by an entrance rule. The behavior of the Kuznetsov process is then studied, which provides insights into trajectory structures of the immigration process. Some well-known results on excessive measures are formulated in terms of stationary immigration processes.