  We construct and study branching Markov processes on the space of finite
configurations of the state space of a given standard process, controlled by a
branching kernel and a killing one. In particular, we may start with a
superprocess, obtaining a branching process with state space the finite
configurations of positive finite measures on a topological space. A main tool
in proving the path regularity of the branching process is the existence of
convenient superharmonic functions having compact level sets, allowing the use
of appropriate potential theoretical methods.
