Summary
Let (ξ s ) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with (ξ s ), which is closely related to the so-called superprocess with spatial motion (ξ s ). In particular, a subsetH of the state space of (ξ s ) intersects the range of the superprocess if and only if the set of paths that hitH is not polar for the path-valued process. The latter property can be investigated 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 (ξ s ) is a diffusion process, we recover certain results recently obtained by Dynkin.
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Le Gall, J.F. A class of path-valued Markov processes and its applications to superprocesses. Probab. Th. Rel. Fields 95, 25–46 (1993). https://doi.org/10.1007/BF01197336
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DOI: https://doi.org/10.1007/BF01197336