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A class of path-valued Markov processes and its applications to superprocesses. (English) Zbl 0794.60076
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.

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.)
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