Continuous dependence of a class of superprocesses on branching parameters and applications

Abstract

A general class of finite variance critical $(\xi, \Phi, k)$-superprocesses $X$ in a Luzin space $E$ with cadlag right Markov motion process $\xi$, regular local branching mechanism and branching functional $k$ of bounded characteristic are shown to continuously depend on $(\Phi, k)$. As an application we show that the processes with a classical branching functional $k(ds) = \varrho_s (\xi_s) ds [that is, a branching functional $k$ generated by a classical branching rate $\varrho_s (y)] are dense in the above class of $(\xi, \Phi, k)$-superprocesses $X$. Moreover, we show that, if the phase space $E$ is a compact metric space and $\xi$ is a Feller process, then always a Hunt version of the $(\xi, \Phi, k)$-superprocess $X$ exists. Moreover, under this assumption, we even get continuity in $(\Phi, k)$ in terms of weak convergence of laws on Skorohod path spaces.