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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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