On the existence of Markovian measures on continuous paths

Abstract

Let η be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space Y, and assume that η admits an invariant measure. We explicit conditions on η such that successive Markovianisation of η over any dense and countable set of times have all limit points (in the weak-star topology on Radon measures) satisfying the strong Markov property. We show that if Y is a locally compact Polish group, and η is left or right translation invariant, then η satisfies such conditions. Our proof uses the Zermalo-Fraenkel and Axiom of Dependant Choice axiomatisation of set theory, in which countable products of compacts are compact.

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BibTeX RIS

BibTeX

@misc{pitcho2026existence,
  title = {{On the existence of Markovian measures on continuous paths}},
  author = {Jules Pitcho},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.29036},
  eprint = {2605.29036},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.29036},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On the existence of Markovian measures on continuous paths
AU  - Jules Pitcho
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2605.29036
UR  - https://arxiv.org/abs/2605.29036
AB  - Let η be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space Y, and assume that η admits an invariant measure. We explicit conditions on η such that successive Markovianisation of η over any dense and countable set of times have all limit points (in the weak-star topology on Radon measures) satisfying the strong Markov property. We show that if Y is a locally compact Polish group, and η is left or right translation invariant, then η satisfies such conditions. Our proof uses the Zermalo-Fraenkel and Axiom of Dependant Choice axiomatisation of set theory, in which countable products of compacts are compact.
ER  -

Abstract

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