Potential-theoretical methods in the construction of measure-valued Markov branching processes

Abstract

Abstract. We develop potential-theoretical methods in the construction of measure-valued branching processes. We complete results of P. J. Fitzsimmons and E. B. Dynkin on the construction, regularity and other properties of the superprocess associated with a given right process and a branching mechanism.

Applications of Compact Superharmonic Functions: Path Regularity and Tightness of Capacities

“…The existence of the compact Lyapunov functions given by Proposition 3.3.1 is the main step in [1] for the proof of the càdlàg property of the paths of the measure-valued (X, Φ)-superprocess, as Proposition 2.2.3 indicates.…”

“…The frame we consider in this subsection is as in [12] and [1]. More precisely, we assume that U is the resolvent of a right Markov process X with state space E, called spatial motion.…”

“…Step III of the proof of Theorem 4.9 in [1] for details. Consider a Ray topology which is finer than the original topology.…”

“…We present now briefly (cf. [12] and [1]) the construction of the measurevalued branching Markov process associated with X and Φ, the (X, Φ)-superprocess, a Borel right process with state space M (E), the space of all positive finite measures on (E, B), endowed with the weak topology.…”

“…This is a main argument in the proof in [1] of the càdlàg property of the paths of the measure-valued process…”