Le Gall, Jean-François A class of path-valued Markov processes and its applications to superprocesses. (English) Zbl 0794.60076 Probab. Theory Relat. Fields 95, No. 1, 25-46 (1993). Let \((\xi_ s)\) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with \((\xi_ s)\), which is closely related to the so-called superprocess with spatial motion \((\xi_ s)\). In particular, a subset \(H\) of the state space of \((\xi_ s)\) intersects the range of the superprocess if and only if the set of paths that hit \(H\) is not polar for the path-valued process. The latter property can be investigated by using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when \((\xi_ s)\) is a diffusion process, we recover certain results recently obtained by Dynkin. Reviewer: J.F.Le Gall (Paris) Cited in 2 ReviewsCited in 34 Documents MSC: 60J45 Probabilistic potential theory 60J25 Continuous-time Markov processes on general state spaces 60G57 Random measures 60J60 Diffusion processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) Keywords:continuous Markov process; superprocess; potential theory; graph of the superprocess; diffusion process PDF BibTeX XML Cite \textit{J.-F. Le Gall}, Probab. Theory Relat. Fields 95, No. 1, 25--46 (1993; Zbl 0794.60076) Full Text: DOI References: [1] Baras, P., Pierre, M.: Probl?mes paraboliques semi-lin?aires avec donn?es mesures. Appl. Anal.18, 111-149 (1984) · Zbl 0582.35060 · doi:10.1080/00036818408839514 [2] Dawson, D.A., Iscoe, I., Perkins, E.A.: Super-Brownian motion: Path properties and hitting probabilities. Probab. Theory Relat. 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